Can the Convergence of a Sequence be Proven by Using Epsilon and Inequalities?

[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)

 

[Dick]

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)

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 $q=1$.

 

[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

 

[dirk_mec1]

You can use the definition in terms of epsilon.