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

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<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>
 
<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>
 
<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>
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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)  \text{ uniformly on } X</math> or <math>f_n \to f \text{ 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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'''Definition''':
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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)  \text{ uniformly on } X</math> or <math>f_n \to f \text{ 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>.
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<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>
  

Revision as of 11:25, 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:

<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" />

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:

<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" />

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

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 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 or ) 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 <math>(f_n(x))_{n=1}^{\infty}<math>:

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