= Solution
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 $r=(m_3,n_3)$. Then
$$
\begin{aligned}
\delta(p,r)&\leq\max\{d_M(m_1,m_2)+d_M(m_2,m_3),\ d_N(n_1,n_2)+d_N(n_2,n_3)\}\\
&\leq\max\{d_M(m_1,m_2),d_N(n_1,n_2)\}+\max\{d_M(m_2,m_3),d_N(n_2,n_3)\}\\
&=\delta(p,q)+\delta(q,r).
\end{aligned}
$$
Therefore \b[the maximum formula defines a <metric>], the <maximum product metric>.
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