Limit Proof Confusion: How does sN+2 cancel with sn-1 in this limit proof?

[MstrGnrl]

Hello Folks,

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

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

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

s_n = s_N*s_N+1/s_N*** s_n/s_n-1

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

Thanks

 

[pbuk]

Can someone please explain this portion of the problem? I don't understand how s_N+2 cancels with s_n-1.

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.

 

[MstrGnrl]

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

 

[pbuk]

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

 

[PeroK]

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

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