I am given the following problem
Let (X; d) be a metric space, and let $\{ y_n\}_{n\in \mathbb{N}}$ and $\{ x_n\}_{n\in \mathbb{N}}$ be two sequences in $(X, d)$, both converging towards $a\in X$. Let $\{ z_n\}_{n\in \mathbb{N}}$ be the sequence defined by $ z_n = \left\{\begin{array}{lc} x_{(n+1)/2} & n \ \text{is odd} \\ y_{n/2} & n \ \text{is even}\end{array} \right..$ Prove that $\{ z_n\}_{n\in \mathbb{N}}$ is Cauchy in $(X,d)$.
I have never quite understood how to prove that sequences is Cauchy, even the definition is a tad blurry for me. I know I need to show that there exists an $\epsilon>0$ such that $n,m \Rightarrow N$ implies $x(n,m)<\epsilon$ for every $N$. But I have problems getting started, every hint or nudge is welcome =)
Sorry for posting homework questions here, but when I do not understand something I want to learn it.