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

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<td><strong>Definition:</strong> An <strong>Infinite Sequence of Functions</strong> <math>(f_n(x))_{n=1}^{\infty} = (f_1(x), f_2(x), ..., f_n(x), ...)</math> is a sequence of functions with a common domain. The <strong><math>n^{\mathrm{th}}</math> Term</strong> of the sequence is the function <math>f_n(x)</math>.</td>
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==Sequences of Functions==
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<blockquote style="background: white; border: 1px solid black; padding: 1em;">  
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'''Definition''':
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An <strong>Infinite Sequence of Functions</strong> <math>(f_n(x))_{n=1}^{\infty} = (f_1(x), f_2(x), ..., f_n(x), ...)</math> is a sequence of functions with a common domain. The <strong><math>n^{\mathrm{th}}</math> Term</strong> of the sequence is the function <math>f_n(x)</math>.
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</blockquote>
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<p><em>We can define a finite sequence of functions analogously. A finite sequence of functions is denoted <math>(f_n)_{n=1}^{\infty}</math>.</em></p>
 
<p><em>We can define a finite sequence of functions analogously. A finite sequence of functions is denoted <math>(f_n)_{n=1}^{\infty}</math>.</em></p>
 
<p><em>We can also denote an infinite sequence of functions as simply <math>(f_n(x))</math>. We can also use curly brackets to denote a sequence of functions such as <math>\{ f_n(x) \}_{n=1}^{\infty}</math> or simply <math>\{ f_n(x) \}</math>.</em></p>
 
<p><em>We can also denote an infinite sequence of functions as simply <math>(f_n(x))</math>. We can also use curly brackets to denote a sequence of functions such as <math>\{ f_n(x) \}_{n=1}^{\infty}</math> or simply <math>\{ f_n(x) \}</math>.</em></p>
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<math>\begin{align} \quad ( f_n(x) )_{n=1}^{\infty} = \left ( nx \right )_{n=1}^{\infty} = (x, 2x, ..., nx, ...) \end{align}</math>
 
<math>\begin{align} \quad ( f_n(x) )_{n=1}^{\infty} = \left ( nx \right )_{n=1}^{\infty} = (x, 2x, ..., nx, ...) \end{align}</math>
 
<p>This is a sequence of diagonal straight lines that pass through the origin and whose slope is increasing. The following illustrates a few of the functions in this sequence:</p>
 
<p>This is a sequence of diagonal straight lines that pass through the origin and whose slope is increasing. The following illustrates a few of the functions in this sequence:</p>
<div class="image-container aligncenter"><img src="http://mathonline.wdfiles.com/local--files/sequences-of-functions/Screen%20Shot%202015-10-19%20at%205.44.29%20PM.png" alt="Screen%20Shot%202015-10-19%20at%205.44.29%20PM.png" class="image" /></div>
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:::[http://mathonline.wdfiles.com/local--files/sequences-of-functions/Screen%20Shot%202015-10-19%20at%205.44.29%20PM.png Graph of a Sequence of Linear Functions]
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<p>For another example, consider the following sequence of functions:</p>
 
<p>For another example, consider the following sequence of functions:</p>
  
 
<math>\begin{align} \quad (f_n(x))_{n=1}^{\infty} = (x^n)_{n=1}^{\infty} = (x, x^2, x^3, ..., x^n, ...) \end{align}</math>
 
<math>\begin{align} \quad (f_n(x))_{n=1}^{\infty} = (x^n)_{n=1}^{\infty} = (x, x^2, x^3, ..., x^n, ...) \end{align}</math>
 
<p>This is a sequence of the simplest <math>n^{\mathrm{th}}</math> degree polynomials whose exponent is increasing. The following illustrates a few of the functions in this sequence:</p>
 
<p>This is a sequence of the simplest <math>n^{\mathrm{th}}</math> degree polynomials whose exponent is increasing. The following illustrates a few of the functions in this sequence:</p>
<div class="image-container aligncenter"><img src="http://mathonline.wdfiles.com/local--files/sequences-of-functions/Screen%20Shot%202015-10-19%20at%205.48.10%20PM.png" alt="Screen%20Shot%202015-10-19%20at%205.48.10%20PM.png" class="image" /></div>
 
  
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:::[http://mathonline.wdfiles.com/local--files/sequences-of-functions/Screen%20Shot%202015-10-19%20at%205.48.10%20PM.png Graph of a Sequence of Polynomials]
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==Pointwise Convergence of Functions==
 
<tr>
 
<tr>
<td><strong>Definition:</strong> Let <math>(f_n(x))_{n=1}^{\infty}</math> be a sequence of functions with common domain <math>X</math>. Then <math>(f_n)_{n=1}^{\infty}</math> is said to be <strong>Pointwise Convergent</strong> to the the function <math>f</math> written <math>\lim_{n \to \infty} f_n(x) = f(x)</math> if for all <math>x \in X</math> and for all <math>\varepsilon > 0</math> there exists a <math>N \in \mathbb{N}</math> such that if <math>n \geq N</math> then <math>\mid f_n(x) - f(x) \mid < \varepsilon</math>.</td>
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<blockquote style="background: white; border: 1px solid black; padding: 1em;">
</tr>
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'''Definition''':  
 +
Let <math>(f_n(x))_{n=1}^{\infty}</math> be a sequence of functions with common domain <math>X</math>. Then <math>(f_n)_{n=1}^{\infty}</math> is said to be <strong>Pointwise Convergent</strong> to the the function <math>f</math> written <math>\lim_{n \to \infty} f_n(x) = f(x)</math> if for all <math>x \in X</math> and for all <math>\varepsilon > 0</math> there exists a <math>N \in \mathbb{N}</math> such that if <math>n \geq N</math> then <math>\mid f_n(x) - f(x) \mid < \varepsilon</math>.
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</blockquote>
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<p>For example, consider the following sequence of functions defined on <math>[0, 1]</math>:</p>
 
<p>For example, consider the following sequence of functions defined on <math>[0, 1]</math>:</p>
  
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<p>To show this, fix <math>x \in [0, 1]</math> and assume that <math>x \neq 0</math> and let <math>\varepsilon > 0</math> be given. Then since <math>n, x > 0</math> we have that:</p>
 
<p>To show this, fix <math>x \in [0, 1]</math> and assume that <math>x \neq 0</math> and let <math>\varepsilon > 0</math> be given. Then since <math>n, x > 0</math> we have that:</p>
  
<math>\begin{align} \quad |f_n(x) - f(x)| = \lvert \frac{1}{n} x - 0 \rvert = \lvert \frac{x}{n} \rvert = \frac{x}{n} \end{align}</math>
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<math>\begin{align} \quad \left|f_n(x) - f(x)\right| = \left|\frac{1}{n} x - 0 \right| = \left|\frac{x}{n} \right| = \frac{x}{n} \end{align}</math>
 
<p>Choose <math>N \in \mathbb{N}</math> such that <math>N > \frac{x}{\varepsilon}</math> which can be done by the Archimedean property. Then <math>\frac{1}{N} < \frac{\varepsilon}{x}</math> and so for <math>n \geq N</math> we have that:</p>
 
<p>Choose <math>N \in \mathbb{N}</math> such that <math>N > \frac{x}{\varepsilon}</math> which can be done by the Archimedean property. Then <math>\frac{1}{N} < \frac{\varepsilon}{x}</math> and so for <math>n \geq N</math> we have that:</p>
  
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<p>This sequence clearly converges to <math>f(0) = 0</math>. So, we conclude that <math>\displaystyle{\lim_{n \to \infty} f_n(x) = f(x)}</math> for all <math>x \in [0, 1]</math>. Hence the sequence <math>(f_n(x))_{n=1}^{\infty}</math> is pointwise convergent on all of <math>[0, 1]</math>.</p>
 
<p>This sequence clearly converges to <math>f(0) = 0</math>. So, we conclude that <math>\displaystyle{\lim_{n \to \infty} f_n(x) = f(x)}</math> for all <math>x \in [0, 1]</math>. Hence the sequence <math>(f_n(x))_{n=1}^{\infty}</math> is pointwise convergent on all of <math>[0, 1]</math>.</p>
  
<h1 id="toc0">Uniform Convergence of Sequences of Functions</h1>
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==Uniform Convergence of Sequences of Functions==
<p>Recall from the <a href="/pointwise-convergence-of-sequences-of-functions">Pointwise Convergence of Sequences of Functions</a> page that we say the sequence of functions <math>(f_n(x))_{n=1}^{\infty}</math> with common domain <math>X</math> is convergent to the limit function <math>f(x)</math> if for all <math>x \in X</math> and for all <math>\varepsilon > 0</math> there exists an <math>N \in \mathbb{N}</math> such that if <math>n \geq N</math> then <math>\mid f_n(x) - f(x) \mid < \varepsilon</math>.</p>
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<p>The sequence of functions <math>(f_n(x))_{n=1}^{\infty}</math> with common domain <math>X</math> is convergent to the limit function <math>f(x)</math> if for all <math>x \in X</math> and for all <math>\varepsilon > 0</math> there exists an <math>N \in \mathbb{N}</math> such that if <math>n \geq N</math> then <math>\mid f_n(x) - f(x) \mid < \varepsilon</math>.</p>
 
<p>Another somewhat stronger type of convergence of a sequence of functions is called uniform convergence which we define below. Note the subtle but very important difference in the definition below!</p>
 
<p>Another somewhat stronger type of convergence of a sequence of functions is called uniform convergence which we define below. Note the subtle but very important difference in the definition below!</p>
<tr>
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<td><strong>Definition:</strong> Let <math>(f_n(x))_{n=1}^{\infty}</math> be a sequence of functions with common domain <math>X</math>. Then <math>(f_n(x))_{n=1}^{\infty}</math> is said to be <strong>Uniformly Convergent</strong> to the the limit function <math>f</math> written <math>\lim_{n \to \infty} f_n(x) = f(x) \: \mathit{uniformly \: on} \: X</math> or <math>f_n \to f \: \mathit{uniformly \: on} \: X</math> if for all <math>\varepsilon > 0</math> there exists a <math>N \in \mathbb{N}</math> such that if <math>n \geq N</math> then <math>\mid f_n(x) - f(x) \mid < \varepsilon</math> for all <math>x \in X</math>.</td>
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<blockquote style="background: white; border: 1px solid black; padding: 1em;">
</tr>
+
'''Definition''':
 +
Let <math>(f_n(x))_{n=1}^{\infty}</math> be a sequence of functions with common domain <math>X</math>. Then <math>(f_n(x))_{n=1}^{\infty}</math> is said to be <strong>Uniformly Convergent</strong> to the the limit function <math>f</math> (written as <math>\lim_{n \to \infty} f_n(x) = f(x)</math> uniformly on <math>X</math> or as <math>f_n \to f </math> uniformly on <math> X</math>) if for all <math>\varepsilon > 0</math> there exists a <math>N \in \mathbb{N}</math> such that if <math>n \geq N</math> then <math>\mid f_n(x) - f(x) \mid < \varepsilon</math> for all <math>x \in X</math>.
 +
</blockquote>
 +
 
 
<p>Graphically, if the sequence of functions <math>(f_n(x))_{n=1}^{\infty}</math> are all real-valued and uniformly converge to the limit function <math>f</math>, then from the definition above, we see that for all <math>\varepsilon > 0</math> there exists an <math>N \in \mathbb{N}</math> such that for all <math>n \geq N</math> we have that the following inequality holds for all <math>x \in X</math>:</p>
 
<p>Graphically, if the sequence of functions <math>(f_n(x))_{n=1}^{\infty}</math> are all real-valued and uniformly converge to the limit function <math>f</math>, then from the definition above, we see that for all <math>\varepsilon > 0</math> there exists an <math>N \in \mathbb{N}</math> such that for all <math>n \geq N</math> we have that the following inequality holds for all <math>x \in X</math>:</p>
  
 
<math>\begin{align} \quad f_n(x) - \varepsilon < f(x) < f_n(x) + \varepsilon \end{align}</math>
 
<math>\begin{align} \quad f_n(x) - \varepsilon < f(x) < f_n(x) + \varepsilon \end{align}</math>
<p>The following graphic illustrates the concept of uniform convergence of a sequence of functions <math>(f_n(x))_{n=1}^{\infty}<math>:</p>
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<p>The following graphic illustrates the concept of uniform convergence of a sequence of functions <math>(f_n(x))_{n=1}^{\infty}</math>:</p>
<div class="image-container aligncenter"><img src="http://mathonline.wdfiles.com/local--files/uniform-convergence-of-sequences-of-functions/Screen%20Shot%202015-10-19%20at%209.25.38%20PM.png" alt="Screen%20Shot%202015-10-19%20at%209.25.38%20PM.png" class="image" /></div>
 
 
 
 
 
  
 +
:::[http://mathonline.wdfiles.com/local--files/uniform-convergence-of-sequences-of-functions/Screen%20Shot%202015-10-19%20at%209.25.38%20PM.png Uniform Convergence of a Sequence of Functions]
  
 
==Licensing==
 
==Licensing==
 
Content obtained and/or adapted from:
 
Content obtained and/or adapted from:
 
* [http://mathonline.wikidot.com/sequences-of-functions Sequences of Functions, mathonline.wikidot.com] under a CC BY-SA license
 
* [http://mathonline.wikidot.com/sequences-of-functions Sequences of Functions, mathonline.wikidot.com] under a CC BY-SA license
* [http://mathonline.wikidot.com/pointwise-convergence-of-sequences-of-functions] under a CC BY-SA license
+
* [http://mathonline.wikidot.com/pointwise-convergence-of-sequences-of-functions Pointwise Convergence of Sequences of Functions] under a CC BY-SA license
* [http://mathonline.wikidot.com/uniform-convergence-of-sequences-of-functions] under a CC BY-SA license
+
* [http://mathonline.wikidot.com/uniform-convergence-of-sequences-of-functions Uniform Convergence of Sequences of Functions] under a CC BY-SA license

Latest revision as of 11:48, 27 October 2021

Sequences of Functions

Definition: An Infinite Sequence of Functions is a sequence of functions with a common domain. The Term of the sequence is the function .

We can define a finite sequence of functions analogously. A finite sequence of functions is denoted .

We can also denote an infinite sequence of functions as simply . We can also use curly brackets to denote a sequence of functions such as or simply .

For example, consider the following sequence of functions:

This is a sequence of diagonal straight lines that pass through the origin and whose slope is increasing. The following illustrates a few of the functions in this sequence:

Graph of a Sequence of Linear Functions

For another example, consider the following sequence of functions:

This is a sequence of the simplest degree polynomials whose exponent is increasing. The following illustrates a few of the functions in this sequence:

Graph of a Sequence of Polynomials

Pointwise Convergence of Functions

Definition: Let be a sequence of functions with common domain . Then is said to be Pointwise Convergent to the the function written if for all and for all there exists a such that if then .

For example, consider the following sequence of functions defined on :

We claim that is pointwise convergent to . The following image shows the first six functions in the sequence given above. It should be intuitively clear that the sequence converges to the limit function .

To show this, fix and assume that and let be given. Then since we have that:

Choose such that which can be done by the Archimedean property. Then and so for we have that:

Therefore for . Now, for , notice that:

This sequence clearly converges to . So, we conclude that for all . Hence the sequence is pointwise convergent on all of .

Uniform Convergence of Sequences of Functions

The sequence of functions with common domain is convergent to the limit function if for all and for all there exists an such that if then .

Another somewhat stronger type of convergence of a sequence of functions is called uniform convergence which we define below. Note the subtle but very important difference in the definition below!

Definition: Let be a sequence of functions with common domain . Then is said to be Uniformly Convergent to the the limit function (written as uniformly on or as uniformly on ) if for all there exists a such that if then for all .

Graphically, if the sequence of functions are all real-valued and uniformly converge to the limit function , then from the definition above, we see that for all there exists an such that for all we have that the following inequality holds for all :

The following graphic illustrates the concept of uniform convergence of a sequence of functions :

Uniform Convergence of a Sequence of Functions

Licensing

Content obtained and/or adapted from: