Real Numbers:Sequences - Revision history

Khanh at 19:21, 11 November 2021

2021-11-11T19:21:26Z

Khanh at 19:18, 11 November 2021

2021-11-11T19:18:42Z

Khanh: /* Proof */

2021-11-06T21:37:49Z

Proof

Khanh at 01:20, 6 November 2021

2021-11-06T01:20:33Z

Lila: /* Contractive Sequence (Definition) */

2021-10-25T22:40:42Z

Contractive Sequence (Definition)

Lila: /* Contractive Sequence (Definition) */

2021-10-25T22:37:13Z

Contractive Sequence (Definition)

Lila: /* Theorem (Cauchy criterion) */

2021-10-25T22:35:24Z

Theorem (Cauchy criterion)

Khanh at 23:22, 15 October 2021

2021-10-15T23:22:31Z

Lila: /* Convergence */

2021-10-13T16:33:06Z

Convergence

Lila: /* Theorem (Convergence of Monotone sequences) */

2021-10-13T16:25:02Z

Theorem (Convergence of Monotone sequences)

@@ Line 330: / Line 330: @@
 :<math>-\epsilon \leq x-x_n\leq \epsilon.\quad\Box</math>
-<h1 id="toc0">The Tail of a Sequence of Real Numbers</h1>
+==The Tail of a Sequence of Real Numbers==
-<p>We will now look at an important aspect of a sequence known as the tail of a sequence.
+We will now look at an important aspect of a sequence known as the tail of a sequence.
 <blockquote style="background: white; border: 1px solid black; padding: 1em;">
@@ Line 364: / Line 364: @@
 This would ''not'' be the ''longest common subsequence'', since <math>Z</math> only has length 3, and the common subsequence <math>\langle B,E,E,B \rangle</math> has length&nbsp;4. The longest common subsequence of <math>X</math> and <math>Y</math> is <math>\langle B,E,G,C,E,B \rangle.</math>
-== Applications ==
+=== Applications ===
 Subsequences have applications to computer science, especially in the discipline of bioinformatics, where computers are used to compare, analyze, and store DNA, RNA, and protein sequences.
@@ Line 389: / Line 389: @@
 Subsequences are used to determine how similar the two strands of DNA are, using the DNA bases: adenine, guanine, cytosine and thymine.
-== Theorems ==
+=== Theorems ===
 * Every infinite sequence of real numbers has an infinite monotone subsequence (This is a lemma used in the proof of the Bolzano–Weierstrass theorem).

@@ Line 329: / Line 329: @@
 :<math>-\epsilon \leq x-x_n\leq \epsilon.\quad\Box</math>
 ==Subsequence==
@@ Line 382: / Line 399: @@
 * [https://en.wikibooks.org/wiki/Real_Analysis/Sequences Sequences, Wikibooks: Real Analysis] under a CC BY-SA license
 * [https://en.wikipedia.org/wiki/Subsequence Subsequence, Wikipedia] under a CC BY-SA license

@@ Line 158: / Line 158: @@
 # If ''x''<sub>n</sub>&nbsp;&le;&nbsp;''y''<sub>n</sub> for every ''n'' in ''N'', then <math>\lim_{n\to\infty}x_n \leq \lim_{n\to\infty}y_n</math>.
 ====Proof====
-. Let ''x''=lim&nbsp;''x''<sub>''n''</sub> and ''y''=lim&nbsp;''y''<sub>''n''</sub>.  We need to show that for any ε&gt;0 there is natural number ''N'' so that if ''n''&ge; ''N'', then |(''x''<sub>''n''</sub>&nbsp;+&nbsp;''y''<sub>''n''</sub>)&nbsp;&minus;&nbsp;(''x''&nbsp;+&nbsp;''y'')|&le;ε. Given any ε&gt;0 we have ε/3&gt;0 so from the [[Real analysis/Sequences#Convergence|definition of convergence]] there is a natural number ''N''<sub>''x''</sub> so that |''x''<sub>''n''</sub>&minus;''x''|&le;ε/3 for all ''n''&gt;''N''<sub>''x''</sub>, similarly we can choose ''N''<sub>''y''</sub> |''y''<sub>''n''</sub>&minus;''y''|&le;ε/3 for all ''n''&gt;''N''<sub>''y''</sub>.
+. Let ''x''=lim&nbsp;''x''<sub>''n''</sub> and ''y''=lim&nbsp;''y''<sub>''n''</sub>.  We need to show that for any ε&gt;0 there is natural number ''N'' so that if ''n''&ge; ''N'', then |(''x''<sub>''n''</sub>&nbsp;+&nbsp;''y''<sub>''n''</sub>)&nbsp;&minus;&nbsp;(''x''&nbsp;+&nbsp;''y'')|&le;ε. Given any ε&gt;0 we have ε/3&gt;0 so from the definition of convergence there is a natural number ''N''<sub>''x''</sub> so that |''x''<sub>''n''</sub>&minus;''x''|&le;ε/3 for all ''n''&gt;''N''<sub>''x''</sub>, similarly we can choose ''N''<sub>''y''</sub> |''y''<sub>''n''</sub>&minus;''y''|&le;ε/3 for all ''n''&gt;''N''<sub>''y''</sub>.
 Let ''N''=max(''N''<sub>x</sub> ,''N''<sub>y</sub>).  If ''n''&gt;''N'', then by the triangle inequality we have

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 * For all integers <math>r</math> and <math>s,</math> every finite sequence of length at least <math>(r - 1)(s - 1) + 1</math> contains a monotonically increasing subsequence of length&nbsp;<math>r</math> ''or'' a monotonically decreasing subsequence of length&nbsp;<math>s</math> (This is the Erdős–Szekeres theorem).
-==Resources==
+== Licensing ==
-* [https://en.wikibooks.org/wiki/Real_Analysis/Sequences Sequences], Wikibooks: Real Analysis
+Content obtained and/or adapted from:

@@ Line 280: / Line 280: @@
 for all <math>n\in\N</math>.
-Every contractive sequence is also a Cauchy sequence (this proof is left to the reader as an exercise), and thus every contractive sequence is convergent by the Cauchy Criterion.
+Every contractive sequence is also a Cauchy sequence (this proof is left to the reader as an exercise), and thus every contractive sequence is convergent by the Cauchy Criterion. However, note that this does not mean that every Cauchy sequence is contractive. The reader is encouraged to find a counterexample to prove this statement (that is, find a sequence that is convergent, and thus Cauchy, that is NOT contractive).
 ==Limit superior and limit inferior==

← Older revision		Revision as of 22:37, 25 October 2021
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	: <math> \|x_{n+2} - x_{n+1}\| \leq k\|x_{n+1} - x_{n}\| </math>		: <math> \|x_{n+2} - x_{n+1}\| \leq k\|x_{n+1} - x_{n}\| </math>
		+
		+	for all <math>n\in\N</math>.
		+
		+	Every contractive sequence is also a Cauchy sequence (this proof is left to the reader as an exercise), and thus every contractive sequence is convergent by the Cauchy Criterion.

	==Limit superior and limit inferior==		==Limit superior and limit inferior==

← Older revision		Revision as of 22:35, 25 October 2021
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	These theorems all describe different aspects of the completeness of the real numbers. The reader will notice that the least upper bound property was used heavily in this section, and it is the axiom that separates the real numbers from the rational numbers. While these theorems would be false for the rational numbers, not all of them can substitute for the least upper bound property. The Cauchy criterion and the nested intervals property are not strong enough to imply the least upper bound property without additional assumptions, while the Convergence of Monotone sequences theorem and the Bolzano—Weierstrass property do imply the least upper bound property.		These theorems all describe different aspects of the completeness of the real numbers. The reader will notice that the least upper bound property was used heavily in this section, and it is the axiom that separates the real numbers from the rational numbers. While these theorems would be false for the rational numbers, not all of them can substitute for the least upper bound property. The Cauchy criterion and the nested intervals property are not strong enough to imply the least upper bound property without additional assumptions, while the Convergence of Monotone sequences theorem and the Bolzano—Weierstrass property do imply the least upper bound property.
		+
		+	===Contractive Sequence (Definition)===
		+	A sequence of real numbers <math> (x_n) </math> is '''contractive''' if there exists some constant <math>k</math>, <math>0 < k < 1</math>, such that
		+
		+	: <math> \|x_{n+2} - x_{n+1}\| \leq k\|x_{n+1} - x_{n}\| </math>

	==Limit superior and limit inferior==		==Limit superior and limit inferior==

@@ Line 81: / Line 81: @@
 |}
-==Convergence==
+==Convergence and Limits==
 A further important property of sequences (arguably the most important property from the perspective of analysis) is the property of convergence. This property can be easily described by extending the epsilon-delta definition. However, because sequences are relative to counting numbers, there exists an additional way to imagine convergence. Both methods are described below.

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 :<math>x- \epsilon < x_N \leq x_n < x</math>.
-Thus |''x''&nbsp;&minus;&nbsp;''x''<sub>''n''</sub>|&nbsp;&lt;&nbsp;ε and by the [[Real analysis/Sequences#Convergence|definition of convergence]], (''x''<sub>''n''</sub>) converges to ''x''. <math>\Box</math>
+Thus |''x''&nbsp;&minus;&nbsp;''x''<sub>''n''</sub>|&nbsp;&lt;&nbsp;ε and by the definition of convergence, (''x''<sub>''n''</sub>) converges to ''x''. <math>\Box</math>
 ===Theorem (Nested intervals property)===