Difference between revisions of "Subsequence/Definition"
From Maths
m (Fixed convergent link) |
(Added another reference, cleaned up, added a note, marked as more cleaning up needed) |
||
| Line 1: | Line 1: | ||
<noinclude> | <noinclude> | ||
| + | {{Requires cleanup|grade=A|msg=The notes are a bit messy}} | ||
==Definition== | ==Definition== | ||
| − | </noinclude>Given a [[sequence]] {{M|1=(x_n)_{n=1}^\infty}} we define a ''subsequence of {{M|1=(x_n)^\infty_{n=1} }}''{{rAPIKM}} as follows: | + | </noinclude>Given a [[sequence]] {{M|1=(x_n)_{n=1}^\infty}} we define a ''subsequence of {{M|1=(x_n)^\infty_{n=1} }}''{{rAPIKM}}{{rFAVIDMH}} as follows: |
| − | * Given any ''strictly'' increasing sequence, {{M|1=(k_n)_{n=1}^\infty}} | + | * Given any ''strictly'' increasing [[monotonic sequence]]<!-- |
| − | ** That means that {{M|\forall n\in\mathbb{N}[k_n<k_{n+1}]}}<ref group="Note">Some books may simply require ''increasing'', this is wrong. Take the theorem from [[Equivalent statements to compactness of a metric space]] which states that a [[metric space]] is [[compact]] {{M|\iff}} every [[sequence]] contains a [[convergent (sequence)|convergent]] subequence. If we only require that: | + | |
| + | START OF FIRST NOTE | ||
| + | |||
| + | --><ref group="Note">Note that ''strictly increasing'' cannot be replaced by ''non-decreasing'' as the sequence could stay the same (ie a term where {{M|m_i\eq m_{i+1} }} for example), it didn't decrease, but it didn't increase either. It must be STRICTLY increasing.<br/><br/>If it was simply "non-decreasing" or just "increasing" then we could define: {{M|k_n:\eq 5}} for all {{M|n}}. | ||
| + | * Then {{M|(x_{k_n})_{n\in\mathbb{N} } }} is a constant sequence where every term is {{M|x_5}} - the 5<sup>th</sup> term of {{M|(x_n)}}.</ref><!-- | ||
| + | |||
| + | END OF FIRST NOTE | ||
| + | |||
| + | -->, {{M|1=(k_n)_{n=1}^\infty\subseteq\mathbb{N} }} | ||
| + | ** That means that {{M|\forall n\in\mathbb{N}[k_n<k_{n+1}]}}<!-- | ||
| + | |||
| + | START OF LONG NOTE | ||
| + | |||
| + | --><ref group="Note">Some books may simply require ''increasing'', this is wrong. Take the theorem from [[Equivalent statements to compactness of a metric space]] which states that a [[metric space]] is [[compact]] {{M|\iff}} every [[sequence]] contains a [[convergent (sequence)|convergent]] subequence. If we only require that: | ||
* {{M|k_n\le k_{n+1} }} | * {{M|k_n\le k_{n+1} }} | ||
Then we can define the sequence: {{M|1=k_n:=1}}. This defines the subsequence {{M|x_1,x_1,x_1,\ldots x_1,\ldots}} of {{M|1=(x_n)_{n=1}^\infty}} which obviously converges. This defeats the purpose of subsequences. | Then we can define the sequence: {{M|1=k_n:=1}}. This defines the subsequence {{M|x_1,x_1,x_1,\ldots x_1,\ldots}} of {{M|1=(x_n)_{n=1}^\infty}} which obviously converges. This defeats the purpose of subsequences. | ||
A subsequence should preserve the "forwardness" of a sequence, that is for a sub-sequence the terms are seen in the same order they would be seen in the parent sequence, and also the "sub" part means building a sequence from it, we want to built a sequence by choosing terms, suggesting we ought not use terms twice. <br/> | A subsequence should preserve the "forwardness" of a sequence, that is for a sub-sequence the terms are seen in the same order they would be seen in the parent sequence, and also the "sub" part means building a sequence from it, we want to built a sequence by choosing terms, suggesting we ought not use terms twice. <br/> | ||
| − | The mapping definition directly supports this, as the mapping can be thought of as choosing terms</ref> | + | The mapping definition directly supports this, as the mapping can be thought of as choosing terms</ref><!-- |
| − | + | ||
| − | * {{M|1=(x_{k_n})_{n=1}^\infty}} | + | END OF LONG NOTE |
| + | |||
| + | --> | ||
| + | Then the subsequence of {{M|(x_n)}} given by {{M|(k_n)}} is: | ||
| + | * {{M|1=(x_{k_n})_{n=1}^\infty}}, the sequence whose terms are: {{M|x_{k_1},x_{k_2},\ldots,x_{k_n},\ldots}} | ||
| + | ** That is to say the {{M|i}}<sup>th</sup> element of {{M|(x_{k_n})}} is the {{M|k_i}}<sup>th</sup> element of {{M|(x_n)}} | ||
===As a mapping=== | ===As a mapping=== | ||
Consider an ([[injection|injective]]) [[mapping]]: {{M|k:\mathbb{N}\rightarrow\mathbb{N} }} with the property that: | Consider an ([[injection|injective]]) [[mapping]]: {{M|k:\mathbb{N}\rightarrow\mathbb{N} }} with the property that: | ||
Latest revision as of 21:54, 16 November 2016
Grade: A
This page requires cleaning up
Some aspect of this page is rather messy and it needs to be cleaned up
The message provided is:
The message provided is:
The notes are a bit messy
Contents
[hide]Definition
Given a sequence (xn)∞n=1 we define a subsequence of (xn)∞n=1[1][2] as follows:
- Given any strictly increasing monotonic sequence[Note 1], (kn)∞n=1⊆N
- That means that ∀n∈N[kn<kn+1][Note 2]
Then the subsequence of (xn) given by (kn) is:
- (xkn)∞n=1, the sequence whose terms are: xk1,xk2,…,xkn,…
- That is to say the ith element of (xkn) is the kith element of (xn)
As a mapping
Consider an (injective) mapping: k:N→N with the property that:
- ∀a,b∈N[a<b⟹k(a)<k(b)]
This defines a sequence, (kn)∞n=1 given by kn:=k(n)
- Now (xkn)∞n=1 is a subsequence
Notes
- Jump up ↑ Note that strictly increasing cannot be replaced by non-decreasing as the sequence could stay the same (ie a term where mi=mi+1 for example), it didn't decrease, but it didn't increase either. It must be STRICTLY increasing.
If it was simply "non-decreasing" or just "increasing" then we could define: kn:=5 for all n.- Then (xkn)n∈N is a constant sequence where every term is x5 - the 5th term of (xn).
- Jump up ↑ Some books may simply require increasing, this is wrong. Take the theorem from Equivalent statements to compactness of a metric space which states that a metric space is compact ⟺ every sequence contains a convergent subequence. If we only require that:
- kn≤kn+1
The mapping definition directly supports this, as the mapping can be thought of as choosing terms
