Prove a sequence S_n converges iff its tail S_n+M converges.

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- Aug 13th 2011, 08:47 AMveronicak5678Tail of a Sequence
Prove a sequence S_n converges iff its tail S_n+M converges.

- Aug 13th 2011, 10:10 AMabhishekkgpRe: Tail of a Sequence
- Aug 13th 2011, 10:45 AMveronicak5678Re: Tail of a Sequence
Yes, that's what I mean. Sorry I don't know Latex!

So... any ideas? - Aug 13th 2011, 07:51 PMabhishekkgpRe: Tail of a Sequence
you just have to use the definition of convergence.

Assume $\displaystyle S_n \rightarrow a$. we will show that $\displaystyle S_{n+M} \rightarrow a$.

$\displaystyle S_n \rightarrow a$ gives us:

$\displaystyle \forall \epsilon >0, \exists N \text{ such that } n>N \Rightarrow |S_n-a|<\epsilon$

To prove $\displaystyle S_{n+M} \rightarrow a$, that is:

$\displaystyle \forall \epsilon>0, \exists N^{'} \text{ such that } n>N^{'} \Rightarrow |S_{n+M}-a|<\epsilon$. We easily see that $\displaystyle N^{'}=N-M$ does the job.

Now can you try and do the 'only if' part. - Aug 13th 2011, 09:45 PMveronicak5678Re: Tail of a Sequence
I see what I was doing wrong. Thanks everyone.