Let $(x_n)$ and $(y_n)$ be sequences such that $\lim y_n = 0$. Suppose that for all $k \in \Bbb N$ and all $m ≥ k$ we have $|x_m − x_k| ≤ y_k$. Show that $(x_n)$ is Cauchy.
I need a little guidance on how to approach the problem. As I see this is the same definition of Cauchy sequences. But I do not see how to connect everything in a logic sequence in order to have a rigorous proof.
My attempt of reasoning
I started first defining the $\lim$ of $y_n$.
For every $\varepsilon>0$ exists $N$ s.t. $n>N$ $|y_n|<\varepsilon$ for all $n>N$ Then I see that all terms of $y_n$ get smaller and smaller as $n$ gets larger. So distance between $x_m$ and $x_k$ gets smaller as the terms get bigger. But one thing that puts me off is that $m ≥ k$ and $| x_m − x_k| ≤ y_k$ why are they $\leq$?
Thanks for help in advance'