If $\{x_n\}$ is a Cauchy sequence, then for all $j \,\, \exists N_j$ such that for $n,m > N_j$ we have $|x_n -x_m| < 2^{-j}$

Question

My uncertainty lies within the following context, from lecture notes

I think it is a pretty bold claim and I am unsure how to derive it. I am grateful if anyone here has a hint.

Answer 1

The definition of Cauchy sequence is this result but changing the $2^{-j}$ with any arbitrary $\epsilon$. Just take $\epsilon = 2^{-j}$ and you are done