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$X$ and $Y$ are compact metric spaces with metrics $d_X$ and $d_Y$. $X \times Y$ is a metric space with the metric $d((x, y),(x , y' )) := \max\{d_x(x, x' ), d_y (y, y' )\}$. I want to show that $X\times Y$ is a compact metric space. I need to use the theorem that totally bounded and complete implies compact, so I need to show all Cauchy sequences converge, but I'm having a difficult time proving it. Any help is appreciated.

Siminore
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1 Answers1

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Let $a_n = (x_n,y_n)$ be a Cauchy sequence in $X \times Y$.

For $\epsilon > 0$, there's $n_0$ for which $n,m \ge n_0 \implies d\left(a_m, a_n \right) < \epsilon$

But $d_X\left(x_n,x_m \right) \le \max\left( d_X \left(x_n,x_m \right), d_Y \left(y_n,y_m \right) \right) = d(a_n,a_m) < \epsilon$.

This shows that $(x_n)$ is Cauchy in $X$, similarly we show that $(y_n)$ is Cauchy in $Y$. Both converge because of completeness and this implies that $(a_n)$ converges.