Let $n \in \mathbb{N}$, and assume $n \geq 1$. For each $i \in \{1,\dots,n\}$, let $(X_i,d_i)$ be a [complete metric space](/page/Complete%20Metric%20Space). Let $X := X_1 \times \cdots \times X_n$. Define the map $d:X \times X \to [0,\infty)$ by
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\begin{align*}
d((x_1,\dots,x_n),(y_1,\dots,y_n)) := \max_{1 \leq i \leq n} d_i(x_i,y_i)
\end{align*}
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for all $(x_1,\dots,x_n),(y_1,\dots,y_n) \in X$. Then $d$ is a metric on $X$, and the [metric space](/page/Metric%20Space) $(X,d)$ is complete.