Given a sequences $(x_{n})\in M$, I will build a new sequences of reals numbers, $t_{mn}=d(x_{m},x_{n})$. Show that $(x_{n})$ is a Cauchy sequences iff, $\displaystyle\lim_{m,n\to\infty}{t_{mn}}=0$. M is a metric space.
My approach: Suppose that, $(x_n)$ is a Cauchy sequence, then give $\epsilon>0$, $\exists N$ such that, for all $n,m\geq N$, $d(x_{n},x_{m})<\epsilon$, and this implies that $t_{mn}<\epsilon$. But I need help with the other direction, I think if $\displaystyle\lim_{n,m\to\infty}{t_{mn}}=\displaystyle\lim_{n,m}{d(x_{m}.,x_{n})}=0$, this mean for every $\epsilon>0$ then exists N, such that for all $m,n\geq N$, $d(x_{m},x_{n})<\epsilon$. Regards!