Cauchy - Sequences. Different definitions

Question

Is $$\forall \epsilon > 0 \exists N \in \mathbb{N}: \forall j \ge N |a_j - a_N| < \epsilon $$ equivalent to

$$\forall \epsilon > 0 \exists N \in \mathbb{N}: \forall m,n \ge N |a_m - a_n| < \epsilon $$

When it comes to cauchy sequences?I Would say yes, because we can choose our N just how we want. And if a sequence gets closer and closer to something, it doesn't matter if we think about the difference between two indexes > N or between $a_N$ and any index >N. We will always find differences that lie in our $\epsilon$. But how would i go on proving it? I understand the thing but i have difficulties proving it mathematicaly..

Thanks for any help.

Answer 1

One direction is simple (just fix $n=N$ and relabel $m=j$). The other direction uses triangle inequality: $$|a_m-a_n| \le |a_m-a_N| + |a_N-a_n| \le 2\epsilon$$ Now relabel $\epsilon >0\Rightarrow 2\epsilon >0$ and obtain the second definition from the first.