Let $(x_n)$ be a sequence of real numbers with the property that for all positive integers $n$ and $m$, $|x_n − x_m|< 1/k$ where $k = \min\{n, m\}$. Prove that the sequence $(x_n)$ is a Cauchy sequence
This is what I have so far:
The definition of a Cauchy sequence is, if given $\epsilon > 0$, there exists and $N$ such that for all $n, m> N$, $|x_n-x_m|< \epsilon$. This is what I have so far for my proof:
Let $\epsilon >0$ be given Pick an $N > 1/\epsilon$, such an $N$ exists by the Archimedean Property of $\mathbb{R}$. This choice of $N$, for any $n,m> N$ we have that $|x_n-x_m|<1/k < 1/n <\epsilon$. I don't fully understand what $k=\min\{n,m\}$ means and how to incorporate this into my proof.
If you note that for all $\epsilon>0$ that there exists an $N \in \mathbb{N}$ so that $\frac{1}{N}<\epsilon$ then it follows for all $m,n \geq N$ that $|x_n-x_m|<\frac{1}{\min(m,n)} \leq \frac{1}{N} < \epsilon$