# Prove the sequence converges

1. Sep 21, 2015

### sebasvargasl

Hello Physics Forums community! I've been struggling for a while with this one ; so basically a sequence a_n is given to us such that the sequence b_n defined by b_n = pa_n + qa_(n+1) is convergent where abs(p)<q. I need to prove a_n is convergent also. Any hint would be of so much help, thank you.

(I've tried proving it is Cauchy but no insight ever comes, just messing with inequalities. I haven't fully understood the role played by abs(p)<q)

2. Sep 21, 2015

### Dick

You can start by trying to think what could go wrong if $|p|<q$ isn't true. Suppose $a_n=(-1)^n$ and $p=1$ and $q=1$.

3. Sep 21, 2015

### sebasvargasl

I'm well aware of the counter examples when the inequality doesn't hold. I just can't find a way to prove the theorem; and to do this I need to fully understand the importance of the inequality. I can't see it intuitively

4. Sep 22, 2015

### dirk_mec1

You can use the definition in terms of epsilon.