Normal core of closed subgroup is closed

Statement
Suppose $$G$$ is a topological group and $$H$$ is a closed subgroup of $$G$$. Then, the normal core of $$H$$ in $$G$$ (i.e., the largest normal subgroup of $$G$$ contained in $$H$$, also defined as the intersection of all the conjugate subgroups of $$H$$) is also a closed subgroup of $$G$$.

Proof
Given: A topological group $$G$$, a closed subgroup $$H$$ of $$G$$.

To prove: The normal core of $$H$$ in $$G$$ is also a closed subgroup of $$G$$.

Proof: