Difference between revisions of "Uniform Convergence of Series of Functions"

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<table class="wiki-content-table">
 
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<td><strong>Theorem 1:</strong> Let <math>(f_n(x))_{n=1}^{\infty}</math> be a sequence of real-valued functions with common domain <math>X</math>. Then <math>\displaystyle{\sum_{n=1}^{\infty} f_n(x)}</math> is uniformly convergent on <math>X</math> if and only if for all <math>\epsilon > 0</math> there exists an <math>N \in \mathbb{N}</math> such that if <math>n \geq N</math> and for all <math>x \in X</math> we have that <math>\displaystyle{ \left| \sum_{k=n+1}^{n+p} f_k(x)  \right| < \epsilon}</math> for all <math>p \in \mathbb{N}</math>.</td>
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<td><strong>Theorem 1:</strong> Let <math>(f_n(x))_{n=1}^{\infty}</math> be a sequence of real-valued functions with common domain <math>X</math>. Then <math>\displaystyle{\sum_{n=1}^{\infty} f_n(x)}</math> is uniformly convergent on <math>X</math> if and only if for all <math>\varepsilon > 0</math> there exists an <math>N \in \mathbb{N}</math> such that if <math>n \geq N</math> and for all <math>x \in X</math> we have that <math>\displaystyle{ \left| \sum_{k=n+1}^{n+p} f_k(x)  \right| < \varepsilon}</math> for all <math>p \in \mathbb{N}</math>.</td>
 
</tr>
 
</tr>
 
</table>
 
</table>
 
<ul>
 
<ul>
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<li><strong>Proof:</strong> <math>\Rightarrow</math> Suppose that <math>\displaystyle{\sum_{n=1}^{\infty} f_n(x)}</math> is uniformly convergent to some limit function <math>f(x)</math> on <math>X</math>. Let <math>(s_n(x))_{n=1}^{\infty}</math> denote the sequence of partial sums for this series. Then we must have that <math>\displaystyle{\lim_{n \to \infty} s_n(x) = f(x)}</math> uniformly on <math>X</math>. So, for <math>\epsilon_1 = \frac{\epsilon}{2}</math> there exists an <math>N \in \mathbb{N}</math> such that if <math>n \geq N</math> and for all <math>x \in X</math> we have that:</li>
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<li><strong>Proof:</strong> <math>\Rightarrow</math> Suppose that <math>\displaystyle{\sum_{n=1}^{\infty} f_n(x)}</math> is uniformly convergent to some limit function <math>f(x)</math> on <math>X</math>. Let <math>(s_n(x))_{n=1}^{\infty}</math> denote the sequence of partial sums for this series. Then we must have that <math>\displaystyle{\lim_{n \to \infty} s_n(x) = f(x)}</math> uniformly on <math>X</math>. So, for <math>\varepsilon_1 = \frac{\varepsilon}{2}</math> there exists an <math>N \in \mathbb{N}</math> such that if <math>n \geq N</math> and for all <math>x \in X</math> we have that:</li>
 
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<div class="math-equation" id="equation-1">\begin{align} \quad \mid s_n(x) - f(x) \mid < \epsilon \end{align}</div>
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<math>\begin{align} \quad \left| s_n(x) - f(x) \right| < \varepsilon \end{align}</math>
 
<ul>
 
<ul>
 
<li>For any <math>p \in \mathbb{N}</math> let <math>m = n + p</math>. Then <math>m \geq N</math> and so:</li>
 
<li>For any <math>p \in \mathbb{N}</math> let <math>m = n + p</math>. Then <math>m \geq N</math> and so:</li>
 
</ul>
 
</ul>
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<div class="math-equation" id="equation-2">\begin{align} \quad \quad  \left| \sum_{k=1}^{m} f_k(x) - \sum_{k=1}^{n} f_k(x)  \right| =  \left| \sum_{k=1}^{n+p} f_k(x) - \sum_{k=1}^{n} f_k(x)  \right| =  \left| \sum_{k=n+1}^{n+p} f_k(x)  \right| = \mid s_m(x) - s_n(x) \mid \leq \mid s_m(x) - f(x) \mid + \mid f(x) - s_n(x) \mid < \epsilon_1 + \epsilon_1 = \frac{\epsilon}{2} + \frac{\epsilon}{2} = \epsilon \end{align}</div>
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<math>\begin{align} \quad \quad  \left| \sum_{k=1}^{m} f_k(x) - \sum_{k=1}^{n} f_k(x)  \right| =  \left| \sum_{k=1}^{n+p} f_k(x) - \sum_{k=1}^{n} f_k(x)  \right| =  \left| \sum_{k=n+1}^{n+p} f_k(x)  \right| = \left| s_m(x) - s_n(x) \right| \leq \left| s_m(x) - f(x) \right| + \left| f(x) - s_n(x) \right| < \varepsilon_1 + \varepsilon_1 = \frac{\varepsilon}{2} + \frac{\varepsilon}{2} = \varepsilon \end{align}</math>
 
<ul>
 
<ul>
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<li><math>\Leftarrow</math> Suppose that for all <math>\epsilon > 0</math> there exists an <math>N \in \mathbb{N}</math> such that if <math>n \geq N</math> and for all <math>x \in X</math> we have that:</li>
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<li><math>\Leftarrow</math> Suppose that for all <math>\varepsilon > 0</math> there exists an <math>N \in \mathbb{N}</math> such that if <math>n \geq N</math> and for all <math>x \in X</math> we have that:</li>
 
</ul>
 
</ul>
  
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<div class="math-equation" id="equation-3">\begin{align} \quad  \left| \sum_{k=n+1}^{n+p} f_k(x)  \right| < \epsilon \end{align}</div>
+
<math>\begin{align} \quad  \left| \sum_{k=n+1}^{n+p} f_k(x)  \right| < \varepsilon \end{align}</math>
 
<ul>
 
<ul>
 
<li>Let <math>m, n \geq N</math>. Assume without loss of generality that <math>m > n</math> and that <math>m = n + p</math> for some <math>p \in \mathbb{N}</math>. Then from above we see that for all <math>x \in X</math>:</li>
 
<li>Let <math>m, n \geq N</math>. Assume without loss of generality that <math>m > n</math> and that <math>m = n + p</math> for some <math>p \in \mathbb{N}</math>. Then from above we see that for all <math>x \in X</math>:</li>
 
</ul>
 
</ul>
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<div class="math-equation" id="equation-4">\begin{align} \quad \mid s_m(x) - s_n(x) \mid =  \left| \sum_{k=1}^{n+p} f_k(x) - \sum_{k=1}^{n} f_k(x)  \right| = \left| \sum_{k=n+1}^{n+p} f_k(x) \right| < \epsilon \end{align}</div>
+
<math>\begin{align} \quad \left| s_m(x) - s_n(x) \right| =  \left| \sum_{k=1}^{n+p} f_k(x) - \sum_{k=1}^{n} f_k(x)  \right| = \left| \sum_{k=n+1}^{n+p} f_k(x) \right| < \varepsilon \end{align}</math>
 
<ul>
 
<ul>
 
<li>So <math>(s_n(x))_{n=1}^{\infty}</math> converges uniformly by the Cauchy uniform convergence criterion for sequences of functions. So <math>\displaystyle{\sum_{n=1}^{\infty} f_n(x)}</math> converges uniformly on <math>X</math>. <math>\blacksquare</math></li>
 
<li>So <math>(s_n(x))_{n=1}^{\infty}</math> converges uniformly by the Cauchy uniform convergence criterion for sequences of functions. So <math>\displaystyle{\sum_{n=1}^{\infty} f_n(x)}</math> converges uniformly on <math>X</math>. <math>\blacksquare</math></li>
 
</ul>
 
</ul>
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==Licensing==
 
==Licensing==
 
Content obtained and/or adapted from:
 
Content obtained and/or adapted from:
 
* [http://mathonline.wikidot.com/pointwise-convergent-and-uniformly-convergent-series-of-func Pointwise Convergent and Uniformly Convergent Series of Functions, mathonline.wikidot.com] under a CC BY-SA license
 
* [http://mathonline.wikidot.com/pointwise-convergent-and-uniformly-convergent-series-of-func Pointwise Convergent and Uniformly Convergent Series of Functions, mathonline.wikidot.com] under a CC BY-SA license

Revision as of 13:51, 27 October 2021

Recall that a sequence of functions 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_n(x))_{n=1}^{\infty}} with common 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 X} is said to be pointwise convergent if for all and 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 \varepsilon > 0} 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 \geq N} then:

Also recall that a sequence of functions 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_n(x))_{n=1}^{\infty}} with common 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 X} is said to be uniformly convergent if 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 \varepsilon > 0} 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 \geq N} then 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 x \in X} 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 \begin{align} \quad \left| f_n(x) - f(x) \right| < \varepsilon \end{align}}

We will now extend the concept of pointwise convergence and uniform convergence to series of functions.

Definition: 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_n(x))_{n=1}^{\infty}} be a sequence of functions with common 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 X} . The corresponding series 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 \displaystyle{\sum_{n=1}^{\infty} f_n(x)}} is said to be Pointwise Convergent to the sum 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(x)} if the corresponding sequence of partial sums 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 (s_n(x))_{n=1}^{\infty}} (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 \displaystyle{s_n(x) = \sum_{k=1}^n f_n(x) = f_1(x) + f_2(x) + ... + f_n(x)}} ) is pointwise convergent 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(x)} .

For example, consider the following sequence of functions defined on the interval 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 (-1, 1)} :

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 \begin{align} \quad (f_n(x))_{n=1}^{\infty} = (x^{n-1})_{n=1}^{\infty} = (1, x, x^2, x^3, ...) \end{align}}

We now that this series converges pointwise 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 x \in (0, 1)} since the result series 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 \sum_{n=1}^{\infty} x^{n-1}} is simply a geometric series to the sum 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 \displaystyle{f(x) = \frac{1}{1 - x}}} .

Definition: 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_n(x))_{n=1}^{\infty}} be a sequence of functions with common 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 X} . The corresponding series 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 \displaystyle{\sum_{n=1}^{\infty} f_n(x)}} is said to be Uniformly Convergent to the sum 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(x)} if the corresponding sequence of partial sums is uniformly convergent 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(x)} .

The geometric series given above actually converges uniformly on 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 (-1, 1)} , though, showing this with the current definition of uniform convergence of series of functions is laborious. We will soon develop methods to determine whether a series of functions converges uniformly or not without having to brute-force apply the definition for uniform convergence for the sequence of partial sums.

Cauchy's Uniform Convergence Criterion for Series of Functions

If we have a sequence of functions 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_n(x))_{n=1}^{\infty}} with common 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 X} then the corresponding series of functions 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 \displaystyle{\sum_{n=1}^{\infty} f_n(x)}} is said to be uniformly convergent if the corresponding sequence of partial sums 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 (s_n(x))_{n=1}^{\infty}} is a uniformly convergent sequence of functions.

We will now look at a nice theorem known as Cauchy's uniform convergence criterion for series of functions.

Theorem 1: 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_n(x))_{n=1}^{\infty}} be a sequence of real-valued functions with common 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 X} . 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 \displaystyle{\sum_{n=1}^{\infty} f_n(x)}} is uniformly convergent on 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} if and only if 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 \varepsilon > 0} 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 \geq N} and 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 x \in X} 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 \displaystyle{ \left| \sum_{k=n+1}^{n+p} f_k(x) \right| < \varepsilon}} 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 p \in \mathbb{N}} .
  • 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 \displaystyle{\sum_{n=1}^{\infty} f_n(x)}} is uniformly convergent to some limit 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(x)} on 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} . 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 (s_n(x))_{n=1}^{\infty}} denote the sequence of partial sums for this series. Then we must 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 \displaystyle{\lim_{n \to \infty} s_n(x) = f(x)}} uniformly on 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} . 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_1 = \frac{\varepsilon}{2}} 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 \geq N} and 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 x \in X} we have that:

  • For any 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 m = n + p} . 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 m \geq N} 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 \begin{align} \quad \quad \left| \sum_{k=1}^{m} f_k(x) - \sum_{k=1}^{n} f_k(x) \right| = \left| \sum_{k=1}^{n+p} f_k(x) - \sum_{k=1}^{n} f_k(x) \right| = \left| \sum_{k=n+1}^{n+p} f_k(x) \right| = \left| s_m(x) - s_n(x) \right| \leq \left| s_m(x) - f(x) \right| + \left| f(x) - s_n(x) \right| < \varepsilon_1 + \varepsilon_1 = \frac{\varepsilon}{2} + \frac{\varepsilon}{2} = \varepsilon \end{align}}

  • 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 \varepsilon > 0} 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 \geq N} and 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 x \in X} 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 \begin{align} \quad \left| \sum_{k=n+1}^{n+p} f_k(x) \right| < \varepsilon \end{align}}

  • 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 m, n \geq N} . Assume without loss of generality 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 m > n} and 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 m = n + p} for some 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 p \in \mathbb{N}} . Then from above we see 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 x \in X} :

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 \begin{align} \quad \left| s_m(x) - s_n(x) \right| = \left| \sum_{k=1}^{n+p} f_k(x) - \sum_{k=1}^{n} f_k(x) \right| = \left| \sum_{k=n+1}^{n+p} f_k(x) \right| < \varepsilon \end{align}}

  • 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 (s_n(x))_{n=1}^{\infty}} converges uniformly by the Cauchy uniform convergence criterion for sequences of functions. 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 \displaystyle{\sum_{n=1}^{\infty} f_n(x)}} converges uniformly on 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} . 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 \blacksquare}

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