# Limit Proof Confusion

1. Sep 16, 2014

### MstrGnrl

Hello Folks,

I am solving a limit proof that has the following attached solution.

Question: Assume all sn ≠ 0. and that the limit L = lim abs(sn+1/sn) exists. Show that if L<1. the lim sn=0

I Understand the solution except for one part which is also attached..

sn = sN*sN+1/sN*** sn/sn-1

Can someone please explain this portion of the problem? I don't understand how sN+2 cancels with sn-1.. It definitely has something to do with n≥N but how they derived it I am not sure

Thanks

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Last edited: Sep 16, 2014
2. Sep 16, 2014

### MrAnchovy

It only cancels when n = N + 3. When n = N + 4 there is another term in the ..... which cancels with the n-1 term, and so on.

3. Sep 16, 2014

### MstrGnrl

Thank you for the response, but I still don't quite understand..

4. Sep 17, 2014

### MrAnchovy

Try writing out the terms for N = 0, n = 3.

5. Sep 17, 2014

### PeroK

Another way to look at it for n > N is to think of n = N + k, then
$$s_n = s_{N+k} = s_N \cdot \frac{s_{N+1}}{s_N} \cdot \frac{s_{N+2}}{s_{N+1}} \dots \frac{s_{N+k-1}}{s_{N+k-2}} \cdot \frac{s_{N+k}}{s_{N+k-1}}$$