Difference between revisions of "Real Function Limits:Sequential Criterion"

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The Sequential Criterion for a Limit of a Function
 
The Sequential Criterion for a Limit of a Function
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<p</math>We will now look at a very important theorem known as The Sequential Criterion for a Limit which merges the concept of the limit of a function <math</math>f</math</math> at a cluster point <math</math>c</math</math> from <math</math>A</math</math> with regards to sequences <math</math>(a_n)</math</math> from <math</math>A</math</math> that converge to <math</math>c</math</math>.</p</math>
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<p>We will now look at a very important theorem known as The Sequential Criterion for a Limit which merges the concept of the limit of a function <math>f</math> at a cluster point <math>c</math> from <math>A</math> with regards to sequences <math>(a_n)</math> from <math>A</math> that converge to <math>c</math>.</p>
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<table class="wiki-content-table"</math>
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<table class="wiki-content-table">
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<tr</math>
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<tr>
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<td</math><strong</math>Theorem 1 (The Sequential Criterion for a Limit of a Function):</strong</math> Let <math</math>f : A \to \mathbb{R}</math</math> be a function and let <math</math>c</math</math> be a cluster point of <math</math>A</math</math>. Then <math</math>\lim_{x \to c} f(x) = L</math</math> if and only if for all sequences <math</math>(a_n)</math</math> from the domain <math</math>A</math</math> where <math</math>a_n \neq c</math</math> <math</math>\forall n \in \mathbb{N}</math</math> and <math</math>\lim_{n \to \infty} a_n = c</math</math> then <math</math>\lim_{n \to \infty} f(a_n) = L</math</math>.</td</math>
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<td><strong>Theorem 1 (The Sequential Criterion for a Limit of a Function):</strong> Let <math>f : A \to \mathbb{R}</math> be a function and let <math>c</math> be a cluster point of <math>A</math>. Then <math>\lim_{x \to c} f(x) = L</math> if and only if for all sequences <math>(a_n)</math> from the domain <math>A</math> where <math>a_n \neq c</math> <math>\forall n \in \mathbb{N}</math> and <math>\lim_{n \to \infty} a_n = c</math> then <math>\lim_{n \to \infty} f(a_n) = L</math>.</td>
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</tr</math>
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</tr>
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</table</math>
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</table>
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<p</math>Consider a function <math</math>f</math</math> that has a limit <math</math>L</math</math> when <math</math>x</math</math> is close to <math</math>c</math</math>. Now consider all sequences <math</math>(a_n)</math</math> from the domain <math</math>A</math</math> where these sequences converge to <math</math>c</math</math>, that is <math</math>\lim_{n \to \infty} a_n = c</math</math>. The Sequential Criterion for a Limit of a Function says that then that as <math</math>n</math</math> goes to infinity, the function <math</math>f</math</math> evaluated at these <math</math>a_n</math</math> will have its limit go to <math</math>L</math</math>.</p</math>
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<p>Consider a function <math>f</math> that has a limit <math>L</math> when <math>x</math> is close to <math>c</math>. Now consider all sequences <math>(a_n)</math> from the domain <math>A</math> where these sequences converge to <math>c</math>, that is <math>\lim_{n \to \infty} a_n = c</math>. The Sequential Criterion for a Limit of a Function says that then that as <math>n</math> goes to infinity, the function <math>f</math> evaluated at these <math>a_n</math> will have its limit go to <math>L</math>.</p>
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<p</math>For example, consider the function <math</math>f: \mathbb{R} \to \mathbb{R}</math</math> defined by the equation <math</math>f(x) = x</math</math>, and suppose we wanted to compute <math</math>\lim_{x \to 0} x</math</math>. We should already know that this limit is zero, that is <math</math>\lim_{x \to 0} x = 0</math</math>. Now consider the sequence <math</math>(a_n) = \left ( \frac{1}{n} \right)</math</math>. This sequence <math</math>(a_n)</math</math> is clearly contained in the domain of <math</math>f</math</math>. Furthermore, this sequence converges to 0, that is <math</math>\lim_{n \to \infty} \frac{1}{n} = 0</math</math>. If all such sequences <math</math>(a_n)</math</math> that converge to <math</math>0</math</math> have the property that <math</math>(f(a_n))</math</math> converges to <math</math>f(0) = 0</math</math>, then we can say that <math</math>\lim_{n \to 0} f(x) = 0</math</math>.</p</math>
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<p>For example, consider the function <math>f: \mathbb{R} \to \mathbb{R}</math> defined by the equation <math>f(x) = x</math>, and suppose we wanted to compute <math>\lim_{x \to 0} x</math>. We should already know that this limit is zero, that is <math>\lim_{x \to 0} x = 0</math>. Now consider the sequence <math>(a_n) = \left ( \frac{1}{n} \right)</math>. This sequence <math>(a_n)</math> is clearly contained in the domain of <math>f</math>. Furthermore, this sequence converges to 0, that is <math>\lim_{n \to \infty} \frac{1}{n} = 0</math>. If all such sequences <math>(a_n)</math> that converge to <math>0</math> have the property that <math>(f(a_n))</math> converges to <math>f(0) = 0</math>, then we can say that <math>\lim_{n \to 0} f(x) = 0</math>.</p>
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<p</math>We will now look at the proof of The Sequential Criterion for a Limit of a Function.</p</math>
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<p>We will now look at the proof of The Sequential Criterion for a Limit of a Function.</p>
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<ul</math>
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<ul>
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<li</math><strong</math>Proof:</strong</math> <math</math>\Rightarrow</math</math> Suppose that <math</math>\lim_{x \to c} f(x) = L</math</math>, and let <math</math>(a_n)</math</math> be a sequence in <math</math>A</math</math> such that <math</math>a_n \neq c</math</math> <math</math>\forall n \in \mathbb{N}</math</math> such that <math</math>\lim_{n \to \infty} a_n = c</math</math>. We thus want to show that <math</math>\lim_{n \to \infty} f(a_n) = L</math</math>.</li</math>
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<li><strong>Proof:</strong> <math>\Rightarrow</math> Suppose that <math>\lim_{x \to c} f(x) = L</math>, and let <math>(a_n)</math> be a sequence in <math>A</math> such that <math>a_n \neq c</math> <math>\forall n \in \mathbb{N}</math> such that <math>\lim_{n \to \infty} a_n = c</math>. We thus want to show that <math>\lim_{n \to \infty} f(a_n) = L</math>.</li>
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</ul</math>
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</ul>
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<ul</math>
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<ul>
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<li</math>Let <math</math>\varepsilon &gt; 0</math</math>. We are given that <math</math>\lim_{x \to c} f(x) = L</math</math> and so for <math</math>\varepsilon &gt; 0</math</math> there exists a <math</math>\delta &gt; 0</math</math> such that if <math</math>x \in A</math</math> and <math</math>0 < \mid x - c \mid < \delta</math</math> then we have that <math</math>\mid f(x) - L \mid < \varepsilon</math</math>. Now since <math</math>\delta &gt; 0</math</math>, since we have that <math</math>\lim_{n \to \infty} a_n = c</math</math> then there exists an <math</math>N \in \mathbb{N}</math</math> such that if <math</math>n ≥ N</math</math> then <math</math>\mid a_n - c \mid < \delta</math</math>. Therefore <math</math>a_n \in V_{\delta} (c) \cap A</math</math>.</li</math>
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<li>Let <math>\varepsilon > 0</math>. We are given that <math>\lim_{x \to c} f(x) = L</math> and so for <math>\varepsilon > 0</math> there exists a <math>\delta > 0</math> such that if <math>x \in A</math> and <math>0 < \mid x - c \mid < \delta</math> then we have that <math>\mid f(x) - L \mid < \varepsilon</math>. Now since <math>\delta > 0</math>, since we have that <math>\lim_{n \to \infty} a_n = c</math> then there exists an <math>N \in \mathbb{N}</math> such that if <math>n ≥ N</math> then <math>\mid a_n - c \mid < \delta</math>. Therefore <math>a_n \in V_{\delta} (c) \cap A</math>.</li>
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</ul</math>
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</ul>
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<ul</math>
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<ul>
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<li</math>Therefore it must be that <math</math>\mid f(a_n) - L \mid < \varepsilon</math</math>, in other words, <math</math>\forall n ≥ N</math</math> we have that <math</math>\mid f(a_n) - L \mid < \varepsilon</math</math> and so <math</math>\lim_{n \to \infty} f(a_n) = L</math</math>.</li</math>
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<li>Therefore it must be that <math>\mid f(a_n) - L \mid < \varepsilon</math>, in other words, <math>\forall n ≥ N</math> we have that <math>\mid f(a_n) - L \mid < \varepsilon</math> and so <math>\lim_{n \to \infty} f(a_n) = L</math>.</li>
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</ul</math>
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</ul>
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<ul</math>
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<ul>
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<li</math><math</math>\Leftarrow</math</math> Suppose that for all <math</math>(a_n)</math</math> in <math</math>A</math</math> such that <math</math>a_n \neq c</math</math> <math</math>\forall n \in \mathbb{N}</math</math> and <math</math>\lim_{n \to \infty} a_n = c</math</math>, we have that <math</math>\lim_{n \to \infty} f(a_n) = L</math</math>. We want to show that <math</math>\lim_{x \to c} f(x) = L</math</math>.</li</math>
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<li><math>\Leftarrow</math> Suppose that for all <math>(a_n)</math> in <math>A</math> such that <math>a_n \neq c</math> <math>\forall n \in \mathbb{N}</math> and <math>\lim_{n \to \infty} a_n = c</math>, we have that <math>\lim_{n \to \infty} f(a_n) = L</math>. We want to show that <math>\lim_{x \to c} f(x) = L</math>.</li>
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</ul</math>
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</ul>
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<ul</math>
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<ul>
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<li</math>Suppose not, in other words, suppose that <math</math>\exists \varepsilon_0 &gt; 0</math</math> such that <math</math>\forall \delta &gt; 0</math</math> then <math</math>\exists x_{\delta} \in A \cap V_{\delta} (c) \setminus \{ c \}</math</math> such that <math</math>\mid f(x_{\delta}) - L \mid ≥ \varepsilon_0</math</math>. Let <math</math>\delta_n = \frac{1}{n}</math</math>. Then there exists <math</math>x_{\delta_n} = a_n \in A \cap V_{\delta_n} (c) \setminus \{ c \}</math</math>, in other words, <math</math>0 < \mid a_n - c \mid < \frac{1}{n}</math</math> and <math</math>\lim_{n \to \infty} a_n = c</math</math>. However, <math</math>\mid f(a_n) - L \mid ≥ \varepsilon_0</math</math> so <math</math>\lim_{n \to \infty} f(a_n) \neq L</math</math>, a contradiction. Therefore <math</math>\lim_{x \to c} f(x) = L</math</math>. <math</math>\blacksquare</math</math></li</math>
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<li>Suppose not, in other words, suppose that <math>\exists \varepsilon_0 > 0</math> such that <math>\forall \delta > 0</math> then <math>\exists x_{\delta} \in A \cap V_{\delta} (c) \setminus \{ c \}</math> such that <math>\mid f(x_{\delta}) - L \mid ≥ \varepsilon_0</math>. Let <math>\delta_n = \frac{1}{n}</math>. Then there exists <math>x_{\delta_n} = a_n \in A \cap V_{\delta_n} (c) \setminus \{ c \}</math>, in other words, <math>0 < \mid a_n - c \mid < \frac{1}{n}</math> and <math>\lim_{n \to \infty} a_n = c</math>. However, <math>\mid f(a_n) - L \mid ≥ \varepsilon_0</math> so <math>\lim_{n \to \infty} f(a_n) \neq L</math>, a contradiction. Therefore <math>\lim_{x \to c} f(x) = L</math>.  
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</ul</math>
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</ul>
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==Resources==
 
==Resources==
 
* [http://mathonline.wikidot.com/the-sequential-criterion-for-a-limit-of-a-function The Sequential Criterion for a Limit of a Function]
 
* [http://mathonline.wikidot.com/the-sequential-criterion-for-a-limit-of-a-function The Sequential Criterion for a Limit of a Function]

Revision as of 09:43, 20 October 2021

The Sequential Criterion for a Limit of a Function

We will now look at a very important theorem known as The Sequential Criterion for a Limit which merges the concept of the limit of a function at a cluster point Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle c} from Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle A} with regards to sequences Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle (a_n)} from Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle A} that converge to Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle c} .

Theorem 1 (The Sequential Criterion for a Limit of a Function): Let Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle f : A \to \mathbb{R}} be a function and let Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle c} be a cluster point of Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle A} . Then Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \lim_{x \to c} f(x) = L} if and only if for all sequences Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle (a_n)} from the domain Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle A} where Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle a_n \neq c} Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \forall n \in \mathbb{N}} and Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \lim_{n \to \infty} a_n = c} then Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \lim_{n \to \infty} f(a_n) = L} .

Consider a function Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle f} that has a limit Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle L} when Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle x} is close to Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle c} . Now consider all sequences Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle (a_n)} from the domain Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle A} where these sequences converge to Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle c} , that is Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \lim_{n \to \infty} a_n = c} . The Sequential Criterion for a Limit of a Function says that then that as Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle n} goes to infinity, the function Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle f} evaluated at these Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle a_n} will have its limit go to Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle L} .

For example, consider the function Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle f: \mathbb{R} \to \mathbb{R}} defined by the equation Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle f(x) = x} , and suppose we wanted to compute Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \lim_{x \to 0} x} . We should already know that this limit is zero, that is Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \lim_{x \to 0} x = 0} . Now consider the sequence Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle (a_n) = \left ( \frac{1}{n} \right)} . This sequence Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle (a_n)} is clearly contained in the domain of Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle f} . Furthermore, this sequence converges to 0, that is Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \lim_{n \to \infty} \frac{1}{n} = 0} . If all such sequences Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle (a_n)} that converge to Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle 0} have the property that Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle (f(a_n))} converges to Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle f(0) = 0} , then we can say that Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \lim_{n \to 0} f(x) = 0} .

We will now look at the proof of The Sequential Criterion for a Limit of a Function.

  • Proof: Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \Rightarrow} Suppose that Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \lim_{x \to c} f(x) = L} , and let Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle (a_n)} be a sequence in Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle A} such that Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle a_n \neq c} Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \forall n \in \mathbb{N}} such that Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \lim_{n \to \infty} a_n = c} . We thus want to show that Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \lim_{n \to \infty} f(a_n) = L} .
  • Let Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \varepsilon > 0} . We are given that Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \lim_{x \to c} f(x) = L} and so for Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \varepsilon > 0} there exists a such that if Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle x \in A} and Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle 0 < \mid x - c \mid < \delta} then we have that Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \mid f(x) - L \mid < \varepsilon} . Now since Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \delta > 0} , since we have that Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \lim_{n \to \infty} a_n = c} then there exists an Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle N \in \mathbb{N}} such that if Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle n ≥ N} then Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \mid a_n - c \mid < \delta} . Therefore Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle a_n \in V_{\delta} (c) \cap A} .
  • Therefore it must be that Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \mid f(a_n) - L \mid < \varepsilon} , in other words, Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \forall n ≥ N} we have that Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \mid f(a_n) - L \mid < \varepsilon} and so Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \lim_{n \to \infty} f(a_n) = L} .
  • Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \Leftarrow} Suppose that for all Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle (a_n)} in Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle A} such that Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle a_n \neq c} Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \forall n \in \mathbb{N}} and Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \lim_{n \to \infty} a_n = c} , we have that Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \lim_{n \to \infty} f(a_n) = L} . We want to show that Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \lim_{x \to c} f(x) = L} .
  • Suppose not, in other words, suppose that Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \exists \varepsilon_0 > 0} such that Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \forall \delta > 0} then Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \exists x_{\delta} \in A \cap V_{\delta} (c) \setminus \{ c \}} such that Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \mid f(x_{\delta}) - L \mid ≥ \varepsilon_0} . Let Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \delta_n = \frac{1}{n}} . Then there exists Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle x_{\delta_n} = a_n \in A \cap V_{\delta_n} (c) \setminus \{ c \}} , in other words, Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle 0 < \mid a_n - c \mid < \frac{1}{n}} and Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \lim_{n \to \infty} a_n = c} . However, Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \mid f(a_n) - L \mid ≥ \varepsilon_0} so Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \lim_{n \to \infty} f(a_n) \neq L} , a contradiction. Therefore Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \lim_{x \to c} f(x) = L} .


Resources

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