= Solution
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 \b[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.
Back to article page
