Q:

Suppose $|x_n-x_k|\le n/k^2$ for all $n$ and $k$. Show that $\{x_n\}$ is Cauchy.

Suppose $|x_n-x_k|\le n/k^2$ for all $n$ and $k$. Show that $\{x_n\}$ is Cauchy.
Attempt: Find $M\in N$ s.t. for $n,k\ge M, n/k^2<\varepsilon \hspace{0.5cm}\forall\varepsilon>0.$
For given $\varepsilon$, let $M>1/\sqrt \varepsilon.$ Then, for $n,k\ge M$ $$|n/k^2|\le n/M^2<n\varepsilon.$$ 
I don't know how to deal with $n$ here. Could you give some hint? 

A:

Hint: Replace $k$ with $n+k$ to get the proper left-hand side of a Cauchy-like inequality.

