Uniform Convergence of Sequences of Functions - Revision history

Lila at 17:48, 27 October 2021

2021-10-27T17:48:23Z

Lila: /* Licensing */

2021-10-27T17:45:28Z

Licensing

Lila at 17:44, 27 October 2021

2021-10-27T17:44:14Z

Lila: /* Uniform Convergence of Sequences of Functions */

2021-10-27T17:42:54Z

Uniform Convergence of Sequences of Functions

Lila: /* Sequences of Functions */

2021-10-27T17:40:33Z

Sequences of Functions

Lila: /* Sequences of Functions */

2021-10-27T17:39:22Z

Sequences of Functions

Lila: /* Pointwise Convergence of Functions */

2021-10-27T17:30:34Z

Pointwise Convergence of Functions

Lila: /* Sequences of Functions */

2021-10-27T17:29:27Z

Sequences of Functions

Lila: /* Uniform Convergence of Sequences of Functions */

2021-10-27T17:28:07Z

Uniform Convergence of Sequences of Functions

Lila: /* Uniform Convergence of Sequences of Functions */

2021-10-27T17:27:08Z

Uniform Convergence of Sequences of Functions

@@ Line 55: / Line 55: @@
 <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>
+<p>The following graphic illustrates the concept of uniform convergence of a sequence of functions <math>(f_n(x))_{n=1}^{\infty}</math>:</p>
 :::[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]

@@ Line 62: / Line 62: @@
 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/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

@@ Line 12: / Line 12: @@
 <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>
-[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]
+:::[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]
 <p>For another example, consider the following sequence of functions:</p>
@@ Line 19: / Line 19: @@
 <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>
-[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]
+:::[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]
 ==Pointwise Convergence of Functions==
@@ Line 57: / Line 57: @@
 <p>The following graphic illustrates the concept of uniform convergence of a sequence of functions <math>(f_n(x))_{n=1}^{\infty}<math>:</p>
-[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]
+:::[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==

@@ Line 44: / Line 44: @@
 ==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>
+<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>
@@ Line 56: / Line 56: @@
 <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>
-<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>
 ==Licensing==

@@ Line 11: / Line 11: @@
 <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>
-<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>
 <p>For another example, consider the following sequence of functions:</p>
@@ Line 17: / Line 19: @@
 <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>
-[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 Functions]
+[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]
 ==Pointwise Convergence of Functions==

@@ Line 16: / Line 16: @@
 <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>
-<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>
 ==Pointwise Convergence of Functions==

@@ Line 20: / Line 20: @@
 ==Pointwise Convergence of Functions==
 <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>
+<blockquote style="background: white; border: 1px solid black; padding: 1em;">
-</tr>
+'''Definition''':
 <p>For example, consider the following sequence of functions defined on <math>[0, 1]</math>:</p>

@@ Line 1: / Line 1: @@
 ==Sequences of Functions==
-<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>
+<blockquote style="background: white; border: 1px solid black; padding: 1em;">
 <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>

@@ Line 39: / Line 39: @@
 <blockquote style="background: white; border: 1px solid black; padding: 1em;">
 '''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 "<math>\lim_{n \to \infty} f_n(x) = f(x)</math> uniformly on <math>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>.
+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>