Closed subset of closed subspace is closed

Statement
Suppose $$X$$ is a topological space and $$B$$ is a closed subset of $$X$$. Suppose $$A$$ is a subset of $$B$$ that is a closed subset of $$B$$ under the subspace topology on $$B$$ from $$X$$. Then, $$A$$ is also a closed subset of $$X$$.

Related facts

 * Open subset of open subspace is open
 * Dense subset of dense subspace is dense