Let $\{x_n\}$ be a sequence of real numbers with the property that for all positive integers $n$ and $m$, $|x_n -x_m|\lt \frac 1{mn}$.
To show $\{x_n\}$ is a cauchy sequence.
Attempt:
A sequence $\{x_n\}$ is said to be cauchy if for each $\epsilon \gt0$ there exist $n_0 \in \mathbb N$ such that $|x_n -x_m|\lt \epsilon $ for all $n,m \ge n_0. $
Now, $|x_n -x_m|\lt \frac 1{mn}\lt \frac 1{n} \lt \epsilon$ if $n\gt \frac{1}{\epsilon} $,
choose $n_0 \gt \frac{1}{\epsilon} $, using Archimedian property and hence we are done .
Am I right ?