Intersection of Closed Set with Compact Subspace is Compact/Proof 1

Proof
Let $\tau_K$ be the subspace topology on $K$.

Let $T_K = \left({K, \tau_K}\right)$ be the topological subspace determined by $K$.

By Closed Set in Topological Subspace, $H \cap K$ is closed in $T_K$.

By Closed Subspace of Compact Space is Compact, $H \cap K$ is compact in $T_K$.

By Subspace Topology does not depend on Ambient Space, $H \cap K$ is compact in $T$.