Proof of A Product of Compact Subsets is Compact in the Product Metric

Let $d_K$ be the restriction of $d_X$ to $K$ and $d_L$ the restriction of $d_Y$ to $L$; these are metrics by claim 1 of The Restriction of a Metric to a Subset Induces the Subspace Topology.

Step 1: the factors are compact topological spaces in their metric topologies. By the definition of a compact subset, the hypothesis on $K$ says that $K$ with the subspace topology inherited from $X$ is a compact topological space. By claim 2 of The Restriction of a Metric to a Subset Induces the Subspace Topology that subspace topology is exactly the collection of subsets of $K$ open in the metric space $(K,d_K)$. Hence $(K,d_K)$, with the topology of its open subsets, is a compact topological space, and likewise for $(L,d_L)$.

Step 2: the product metric of the restrictions is the restriction of the product metric. Let $(x_1,y_1),(x_2,y_2)\in K\times L$. By the definition of the product metric, the product metric of $d_K$ and $d_L$ takes at this pair of points the value $\max\{d_K(x_1,x_2),d_L(y_1,y_2)\}=\max\{d_X(x_1,x_2),d_Y(y_1,y_2)\}$, which is the value of $d_{X\times Y}$, hence of $d_{K\times L}$, at the same pair. So the product metric of $d_K$ and $d_L$ equals $d_{K\times L}$.

Step 3: compactness of the product. By Product of Two Compact Spaces is Compact applied to the two compact topological spaces of Step 1, the set $K\times L$ equipped with the product topology of those two topologies is compact. By The Product Metric Induces the Product Topology applied to the metric spaces $(K,d_K)$ and $(L,d_L)$, that product topology is exactly the collection of subsets of $K\times L$ open in the metric space $(K\times L,d_{K\times L})$, using Step 2. This proves the second assertion of the corollary.

Step 4: compactness as a subset of $X\times Y$. By claim 2 of The Restriction of a Metric to a Subset Induces the Subspace Topology, applied to the metric space $(X\times Y,d_{X\times Y})$ and its subset $K\times L$, the collection of subsets of $K\times L$ open in $(K\times L,d_{K\times L})$ is the subspace topology inherited from $X\times Y$. By Step 3 that topology is compact, so $K\times L$ is compact in $X\times Y$ in the sense of the definition of a compact subset.