Let $\{x_n\}$ be a sequence of real numbers with $|x_n − x_m| < 1/(n+m)$ where $n,m$ are positive integers. . k  Prove that the sequence $\{x_n \}$ is Cauchy.
Here is what I have so far..
Let $\epsilon >0$ be given. There exists $N> 1/\epsilon$ by the Archimedean property.
For any $n, m≥N$
$|x_n-x_m|< 1/(n+m) < 1/N < 1/(1/\epsilon)) = \epsilon$
This proves the sequence is Cauchy. I am not sure my proof is correct when I jump from $1/(n+m) <1/N...=\epsilon$.