Convergent Sequences in Metric Spaces - Revision history

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The Boundedness of Convergent Sequences in Metric Spaces

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The Boundedness of Convergent Sequences in Metric Spaces

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The Boundedness of Convergent Sequences in Metric Spaces

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The Boundedness of Convergent Sequences in Metric Spaces

Lila at 16:50, 8 November 2021

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Lila: Created page with "===Limits of Sequences in Metric Spaces===
Recall that if a sequence of real numbers $(x_n)_{n=1}^{\infty} = (x_1, x_2, ..., x_n, ...)$

2021-11-08T16:40:36Z

Created page with "===Limits of Sequences in Metric Spaces=== Recall that if a sequence of real numbers <math>(x_n)_{n=1}^{\infty} = (x_1, x_2, ..., x_n, ...)</math>..."

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===Limits of Sequences in Metric Spaces===
Recall that if a sequence of real numbers <math>(x_n)_{n=1}^{\infty} = (x_1, x_2, ..., x_n, ...)</math> is an infinite ordered list where <math>x_k \in \mathbb{R}</math> for every <math>k \in \{ 1, 2, ... \}</math>. We will now generalize the concept of a sequence to contain elements from a metric space <math>(M, d)</math>.
<blockquote style="background: white; border: 1px solid black; padding: 1em;">
<td>Definition: Let <math>(M, d)</math> be a metric space. An (infinite) Sequence in <math>M</math> denoted <math>(x_n)_{n=1}^{\infty} = (x_1, x_2, ..., x_n, ...)</math> is an infinite ordered list of elements <math>x_k \in M</math> for all <math>k \in \{1, 2, ... \}</math>.</td>
</blockquote>
Finite sequences in a metric space can be defined as a finite ordered list of elements in <math>M</math> but their study is not that interesting to us.
We can also define whether a sequence <math>(x_n)_{n=1}^{\infty}</math> of elements from a metric space <math>(M, d)</math> converges or diverges.
<blockquote style="background: white; border: 1px solid black; padding: 1em;">
<td>Definition: Let <math>(M, d)</math> be a metric space. A sequence <math>(x_n)_{n=1}^{\infty}</math> in <math>M</math> is said to be Convergent to the element <math>p \in M</math> written <math>\lim_{n \to \infty} x_n = p</math> if <math>\lim_{n \to \infty} d(x_n, p) = 0</math> and the element <math>p</math> is said to be the Limit of the sequence <math>(x_n)_{n=1}^{\infty}</math>. If no such <math>p \in M</math> exists, then <math>(x_n)_{n=1}^{\infty}</math> is said to be Divergent.</td>
</blockquote>
There is a subtle but important point to make. In the definition above, <math>\lim_{n \to \infty} x_n = p</math> represents the limit of a sequence of elements from the metric space <math>(M,d)</math> to an element <math>p \in M</math> while <math>\lim_{n \to \infty} d(x_n, p) = 0</math> represents the limit of a sequence of positive real numbers to <math>0</math> - such limits we already have experience with.
<div class="image-container aligncenter"><img src="http://mathonline.wdfiles.com/local--files/limits-of-sequences-in-metric-spaces/Screen%20Shot%202015-09-24%20at%2011.56.32%20AM.png" alt="Screen%20Shot%202015-09-24%20at%2011.56.32%20AM.png" class="image" /></div>
For example, if <math>M</math> is any nonempty set, <math>d : M \times M \to [0, \infty)</math> is the discrete metric, and <math>x \in M</math>, then the sequence defined by <math>x_n = x</math> for all <math>n \in \{ 1, 2, ... \}</math>, then the sequence:

<div style="text-align: center;"><math>\begin{align} \quad (x_n)_{n=1}^{\infty} = (x)_{n=1}^{\infty} = (x, x, ..., x, ...) \end{align}</math></div>
Furthermore, it's not hard to see that this sequence converges to <math>x</math>, i.e., <math>\lim_{n \to \infty} x_n = x</math>, i.e., <math>\lim_{n \to \infty} d(x_n, x) = 0</math> since for all <math>x_n</math> we have that <math>d(x_n, x) = 0</math>, so <math>\lim_{n \to \infty} d(x_n, x) = \lim_{n \to \infty} 0 = 0</math>.
We will soon see that many of theorems regarding limits of sequences of real numbers are analogous to limits of sequences of elements from metric spaces.

← Older revision		Revision as of 17:00, 8 November 2021
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	* [http://mathonline.wikidot.com/limits-of-sequences-in-metric-spaces Limits of Sequences in Metric Spaces, mathonline.wikidot.com] under a CC BY-SA license		* [http://mathonline.wikidot.com/limits-of-sequences-in-metric-spaces Limits of Sequences in Metric Spaces, mathonline.wikidot.com] under a CC BY-SA license
	* [http://mathonline.wikidot.com/the-boundedness-of-convergent-sequences-in-metric-spaces The Boundedness of Convergent Sequences in Metric Spaces, mathonline.wikidot.com] under a CC BY-SA license		* [http://mathonline.wikidot.com/the-boundedness-of-convergent-sequences-in-metric-spaces The Boundedness of Convergent Sequences in Metric Spaces, mathonline.wikidot.com] under a CC BY-SA license
		+	* [http://mathonline.wikidot.com/convergent-sequences-and-subsequences-in-metric-spaces Convergent Sequences and Subsequences in Metric Spaces, mathonline.wikidot.com] under a CC BY-SA license

@@ Line 47: / Line 47: @@
 <ul>
 <li>Therefore <span class="math-inline"><math>\{ x_1, x_2, ..., x_n, ... \}</math></span> is a bounded set in <span class="math-inline"><math>M</math></span>. <span class="math-inline"><math>\blacksquare</math></span></li>
 </ul>

@@ Line 47: / Line 47: @@
 <ul>
 <li>Therefore <span class="math-inline"><math>\{ x_1, x_2, ..., x_n, ... \}</math></span> is a bounded set in <span class="math-inline"><math>M</math></span>. <span class="math-inline"><math>\blacksquare</math></span></li>
 </ul>

@@ Line 47: / Line 47: @@
 <ul>
 <li>Therefore <span class="math-inline"><math>\{ x_1, x_2, ..., x_n, ... \}</math></span> is a bounded set in <span class="math-inline"><math>M</math></span>. <span class="math-inline"><math>\blacksquare</math></span></li>
 </ul>

← Older revision		Revision as of 16:52, 8 November 2021
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	Content obtained and/or adapted from:		Content obtained and/or adapted from:
	* [http://mathonline.wikidot.com/limits-of-sequences-in-metric-spaces Limits of Sequences in Metric Spaces, mathonline.wikidot.com] under a CC BY-SA license		* [http://mathonline.wikidot.com/limits-of-sequences-in-metric-spaces Limits of Sequences in Metric Spaces, mathonline.wikidot.com] under a CC BY-SA license
		+	* [http://mathonline.wikidot.com/the-boundedness-of-convergent-sequences-in-metric-spaces The Boundedness of Convergent Sequences in Metric Spaces, mathonline.wikidot.com] under a CC BY-SA license

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 <p>If <span class="math-inline"><math>(M, d)</math></span> is a metric space and <span class="math-inline"><math>(x_n)_{n=1}^{\infty}</math></span> is a sequence in <span class="math-inline"><math>M</math></span> that is convergent then the limit of this sequence <span class="math-inline"><math>p \in M</math></span> is unique.</p>
 <p>We will now look at another rather nice theorem which states that if <span class="math-inline"><math>(x_n)_{n=1}^{\infty}</math></span> is convergent then it is also bounded.</p>
-<table class="wiki-content-table">
+<blockquote style="background: white; border: 1px solid black; padding: 1em;">
 <td><strong>Theorem 1:</strong> Let <span class="math-inline"><math>(M, d)</math></span> be a metric space and let <span class="math-inline"><math>(x_n)_{n=1}^{\infty}</math></span> be a sequence in <span class="math-inline"><math>M</math></span>. If <span class="math-inline"><math>(x_n)_{n=1}^{\infty}</math></span> is convergent then the set <span class="math-inline"><math>\{ x_1, x_2, ..., x_n, ... \}</math></span> is bounded.</td>
-</tr>
+</blockquote>
 <ul>
 <li><strong>Proof:</strong> Let <span class="math-inline"><math>(M, d)</math></span> be a metric space and let <span class="math-inline"><math>(x_n)_{n=1}^{\infty}</math></span> be a sequence in <span class="math-inline"><math>M</math></span> that converges to <span class="math-inline"><math>p \in M</math></span>, i.e., <span class="math-inline"><math>\lim_{n \to \infty} x_n = p</math></span>. Then <span class="math-inline"><math>\lim_{n \to \infty} d(x_n, p) = 0</math></span>. So for all <span class="math-inline"><math>\epsilon > 0</math></span> there exists an <span class="math-inline"><math>N \in \mathbb{N}</math></span> such that if <span class="math-inline"><math>n \geq N</math></span> then <span class="math-inline"><math>d(x_n, p) < \epsilon</math></span>. So for <span class="math-inline"><math>\epsilon_0 = 1 > 0</math></span> there exists an <span class="math-inline"><math>N(\epsilon_0) \in \mathbb{N}</math></span> such that if <span class="math-inline"><math>n \geq N(\epsilon_0)</math></span> then:</li>

@@ Line 18: / Line 18: @@
 <p>Furthermore, it's not hard to see that this sequence converges to <span class="math-inline"><math>x</math></span>, i.e., <span class="math-inline"><math>\lim_{n \to \infty} x_n = x</math></span>, i.e., <span class="math-inline"><math>\lim_{n \to \infty} d(x_n, x) = 0</math></span> since for all <span class="math-inline"><math>x_n</math></span> we have that <span class="math-inline"><math>d(x_n, x) = 0</math></span>, so <span class="math-inline"><math>\lim_{n \to \infty} d(x_n, x) = \lim_{n \to \infty} 0 = 0</math></span>.</p>
 <p>We will soon see that many of theorems regarding limits of sequences of real numbers are analogous to limits of sequences of elements from metric spaces.</p>
 == Licensing ==
 Content obtained and/or adapted from:
 * [http://mathonline.wikidot.com/limits-of-sequences-in-metric-spaces Limits of Sequences in Metric Spaces, mathonline.wikidot.com] under a CC BY-SA license

← Older revision		Revision as of 16:43, 8 November 2021
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	<p>Furthermore, it's not hard to see that this sequence converges to <span class="math-inline"><math>x</math></span>, i.e., <span class="math-inline"><math>\lim_{n \to \infty} x_n = x</math></span>, i.e., <span class="math-inline"><math>\lim_{n \to \infty} d(x_n, x) = 0</math></span> since for all <span class="math-inline"><math>x_n</math></span> we have that <span class="math-inline"><math>d(x_n, x) = 0</math></span>, so <span class="math-inline"><math>\lim_{n \to \infty} d(x_n, x) = \lim_{n \to \infty} 0 = 0</math></span>.</p>		<p>Furthermore, it's not hard to see that this sequence converges to <span class="math-inline"><math>x</math></span>, i.e., <span class="math-inline"><math>\lim_{n \to \infty} x_n = x</math></span>, i.e., <span class="math-inline"><math>\lim_{n \to \infty} d(x_n, x) = 0</math></span> since for all <span class="math-inline"><math>x_n</math></span> we have that <span class="math-inline"><math>d(x_n, x) = 0</math></span>, so <span class="math-inline"><math>\lim_{n \to \infty} d(x_n, x) = \lim_{n \to \infty} 0 = 0</math></span>.</p>
	<p>We will soon see that many of theorems regarding limits of sequences of real numbers are analogous to limits of sequences of elements from metric spaces.</p>		<p>We will soon see that many of theorems regarding limits of sequences of real numbers are analogous to limits of sequences of elements from metric spaces.</p>
		+
		+	== Licensing ==
		+	Content obtained and/or adapted from:
		+	* [http://mathonline.wikidot.com/limits-of-sequences-in-metric-spaces Limits of Sequences in Metric Spaces, mathonline.wikidot.com] under a CC BY-SA license

@@ Line 10: / Line 10: @@
 </blockquote>
 <p><em>There is a subtle but important point to make. In the definition above, <span class="math-inline"><math>\lim_{n \to \infty} x_n = p</math></span> represents the limit of a sequence of elements from the metric space <span class="math-inline"><math>(M,d)</math></span> to an element <span class="math-inline"><math>p \in M</math></span> while <span class="math-inline"><math>\lim_{n \to \infty} d(x_n, p) = 0</math></span> represents the limit of a sequence of positive real numbers to <span class="math-inline"><math>0</math></span> - such limits we already have experience with.</em></p>
-<div class="image-container aligncenter"><img src="http://mathonline.wdfiles.com/local--files/limits-of-sequences-in-metric-spaces/Screen%20Shot%202015-09-24%20at%2011.56.32%20AM.png" alt="Screen%20Shot%202015-09-24%20at%2011.56.32%20AM.png" class="image" /></div>
 <p>For example, if <span class="math-inline"><math>M</math></span> is any nonempty set, <span class="math-inline"><math>d : M \times M \to [0, \infty)</math></span> is the discrete metric, and <span class="math-inline"><math>x \in M</math></span>, then the sequence defined by <span class="math-inline"><math>x_n = x</math></span> for all <span class="math-inline"><math>n \in \{ 1, 2, ... \}</math></span>, then the sequence:</p>

Convergent Sequences in Metric Spaces - Revision history

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Lila: /* The Boundedness of Convergent Sequences in Metric Spaces */

Lila: /* The Boundedness of Convergent Sequences in Metric Spaces */

Lila: /* Licensing */

Lila: /* The Boundedness of Convergent Sequences in Metric Spaces */

Lila at 16:50, 8 November 2021

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Lila: Created page with "===Limits of Sequences in Metric Spaces=== Recall that if a sequence of real numbers (x_n)_{n=1}^{\infty} = (x_1, x_2, ..., x_n, ...)..."

Lila: Created page with "===Limits of Sequences in Metric Spaces===
Recall that if a sequence of real numbers $(x_n)_{n=1}^{\infty} = (x_1, x_2, ..., x_n, ...)$