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: '''Unnamed Theorem Lemma or Corollary: 1'''
{{Requires work|grade=A*|msg=This needs a "still unnamed" classification, so one has a category of unnamed theorems, as this page will persist even if it is given a name}}
__TOC__
==Statement==
Let {{M|(a_n)_{n\in\mathbb{N} } \subset\mathbb{R} }} be a ''[[real numbers|real]]'' [[sequence]]. Suppose {{M|\lim_{n\rightarrow\infty}(a_n)\eq\ell}}. Then:
* {{M|(b_n)_{n\in\mathbb{N} }\subset\mathbb{R} }} defined by: {{M|1=b_n:\eq\left\{\begin{array}{lr}a_{n-1}&\text{if }n\ge 2\\c\in\mathbb{R}&\text{if }n\eq 1\end{array}\right.}} also {{link|converges|sequence}} to {{M|\ell}}<ref name="Alec">Alec's own work. Intermediate lemma used in [[If a real series converges then the limit of the terms of the series is zero]]</ref>.
** Here {{M|c\in\mathbb{R} }} is any value. I used {{M|0\in\mathbb{R} }} on paper as I was working with [[series (summation)]] and this could be used to show {{M|\lim_{n\rightarrow\infty}(\sum^n_{i\eq 1}a_n)\eq\lim_{n\rightarrow\infty}(\sum^n_{i\eq 1}b_n)}} as {{M|1=\sum_{i=1}^n b_n\eq 0+\sum_{i=2}^n b_n=0+\sum_{i=1}^{n-1}a_n}}
*** Which is to show: {{M|\lim_{n\rightarrow\infty}(S_n)\eq\lim_{n\rightarrow\infty}(S_{n-1})\eq\ell}} - a vital part in [[If a real series converges then the limit of the terms of the series is zero]]
==Proof==
Suppose that {{M|\lim_{n\rightarrow\infty}(a_n)\eq\ell}}, and {{M|(b_n)_{n\in\mathbb{N} } }} is defined as above, for an arbitrary constant {{M|c\in\mathbb{R} }}. We wish to show that:
* {{M|\lim_{n\rightarrow\infty}(b_n)\eq\ell}} which can be stated symbolically: {{M|1=\forall\epsilon>0\exists N\in\mathbb{N}\forall n\in\mathbb{N}[n>N\implies d(b_n,\ell)<\epsilon]}}
'''Proof:'''
* Let {{M|\epsilon>0}} be given.
** Choose {{M|N:\eq N'+1}} where {{M|N'\in\mathbb{N} }} exists by hypothesis and means: {{M|1=\forall n'\in\mathbb{N}[n'>N'\implies d(a_n,\ell)<\epsilon]}}
*** Let {{M|n\in\mathbb{N} }} be given
**** Suppose {{M|n\le N}} - by the nature of [[logical implication]] we do not care whether or not {{M|d(b_n,\ell)<\epsilon}} the implication is true regardless.
**** Suppose {{M|n > N}}
***** This means {{M|n>N'+1}} by definition of {{M|N}}, so {{M|n-1>N'}}
****** This means {{M|d(a_{n-1},\ell)<\epsilon}} (by hypothesis and what {{M|N'}} is defined to be)
***** {{Caution|1=The reader must be sure {{M|a_{n-1} }} exists. I.E that {{M|n-1\ge 1}} or {{M|n\ge 2}}. As {{M|N'\in\mathbb{N} }} we see {{M|N'\ge 1}}, then {{M|N:\eq N'+1}} so {{M|N'+1\ge 2}} giving {{M|N\ge 2}}. Then we have {{M|n>N\ge 2}} giving {{M|n> 2}} or {{M|n\ge 3}}. As {{M|n}} is at least 3, we can be sure {{M|a_{n-1} }} exists.}}
****** {{Warning|1=This varies slightly depending on your definition of [[convergent sequence]], if we had had: {{M|\forall\epsilon>0\exists N\in\mathbb{N}\forall n\in\mathbb{N}[n\ge N\implies d(x_n,x)<\epsilon]}} - notice the {{M|n\ge N}} rather than our {{M|n>N}} we would have to consider this differently}}
****** Ultimately all of this can be avoided by saying {{M|N:\eq\max(N'+1,2)}} 
****** We also probably already have this from the definition of {{M|N'}} - I didn't check. My sanity check is in the "caution" above.
***** We now know {{M|n\ge 3}} and {{M|d(a_{n-1},\ell)<\epsilon}}. Note that for such an {{M|n}} we have:
****** {{M|b_n:\eq a_{n-1} }}
***** Thus we see {{M|d(b_n,\ell)<\epsilon}} as required.
==Possible generalisation==
I almost created the page:
* ''If a sequence converges then all sequences that are eventually equal to it converge also''
Where "eventually equal"

The statement I have written (but scrapped, in favour of this page) is included below.
===Statement===
Let {{M|(a_n)_{n\in\mathbb{N} } }} be a [[convergent sequence]] in a [[metric space]] {{M|(X,d)}}. Then any sequence {{link|eventually|sequence}} equal to it, {{M|(b_n)_{n\in\mathbb{N} } }} converges also. Furthermore {{M|(b_n)}} converges to the same limit as {{M|(a_n)}}

That is to say {{M|(b_n)_{n\in\mathbb{N} } }} is a sequence such that:
* {{M|\exists M\in\mathbb{N}\forall m\in\mathbb{N}[a_m\eq b_{m+M}]}}
Then {{M|(b_n)}} converges to the same limit as {{M|(a_n)}} (should {{M|(a_n)}} converge)
==References==
<references/>
{{Theorem Of|Metric Space|Analysis|Real Analysis|Functional Analysis}}