= Solution
Each coordinate distance is at most the maximum, so convergence of the pairs implies convergence of both coordinates. Conversely, for any $\epsilon>0$, coordinate convergence gives indices $N_M,N_N$ after which the respective distances are below $\epsilon$. Beyond their maximum, both are below $\epsilon$, and so is the <maximum product metric>.
$$
\boxed{(x_n,y_n)\to(x,y)\iff x_n\to x\text{ and }y_n\to y.}
$$
The same distance comparison will also handle <Cauchy sequences>.
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