Converges of two different sequences using Cauchy

Question

I've recently started calc 1. I've had this problem :

let $a_n$ and $b_n$ be two sequences such that $\forall\varepsilon$ $\exists N$ $\in\mathbb{N}$ such that for $\forall m,n > N$ $|a_n - b_m| < \varepsilon $

prove that both $a_n$ and $b_n$ converge to the same limit.

I've been sitting on this one for 3 days now , I can't figure out how to break them up to two different sequences, I've tried to build a new sequence such that $c_n = a_n - b_m$ but didn't have much luck with continuing from there, I think that you need to prove that both sequences are Cauchy sequences and from there you'll be able to conclude that they have the same limit but aging I'm not sure how to get there.

any help would be much appreciated

Answer 1

There is some $N$ s.t. $\forall n,m >N$ we have $$\vert a_{n}-b_{m}\vert <\frac{\epsilon }{2}$$ It follows that $\forall n,m,l >N$ $$\vert b_{m}-b_{l}\vert \leq \vert b_{m}-a_{n}\vert +\vert b_{l}-a_{n}\vert < \epsilon$$... can you take it from there?

Edit: Once you have that both sequences are Cauchy, I would switch over to a proof by contradiction.