# Thread: Tail of a Sequence

1. ## Tail of a Sequence

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

2. ## Re: Tail of a Sequence

Originally Posted by veronicak5678
Prove a sequence S_n converges iff its tail S_n+M converges.
do you mean $\displaystyle S_n$ converges iff $\displaystyle S_{n+M}$ converges?

3. ## Re: Tail of a Sequence

Yes, that's what I mean. Sorry I don't know Latex!
So... any ideas?

4. ## Re: Tail of a Sequence

Originally Posted by veronicak5678
Yes, that's what I mean. Sorry I don't know Latex!
So... any ideas?
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.

5. ## Re: Tail of a Sequence

I see what I was doing wrong. Thanks everyone.