Past exam of the metric spaces and complex analysis course of the University of Oxford / 2014 / 2 / a

Nonnegativity and symmetry follow from the corresponding metric axioms in the two factors. The maximum is zero exactly when both coordinate distances are zero, hence exactly when the ordered pairs are equal.

For the triangle inequality, put $p=(m_1,n_1)$, $q=(m_2,n_2)$ and $r=(m_3,n_3)$. ThenTherefore the maximum formula defines a metric, the maximum product metric.

Each coordinate distance is at most the maximum, so convergence of the pairs implies convergence of both coordinates. Conversely, for any $ϵ>0$, coordinate convergence gives indices $N_M,N_N$ after which the respective distances are below $ϵ$. Beyond their maximum, both are below $ϵ$, and so is the maximum product metric.The same distance comparison will also handle Cauchy sequences.

Let $(x_n,y_n)$ be a Cauchy sequence in the product. Since each coordinate distance is bounded by $\delta$, both $(x_n)$ and $(y_n)$ are Cauchy in their respective factors. By completeness they have limits $x\in M$, $y\in N$. Part (ii) then gives convergence of the pairs to $(x,y)$.

Thus the product of two complete metric spaces is complete under the maximum product metric. This checks every Cauchy sequence, rather than assuming beforehand that it has a coordinate limit.