Dense subset of dense subspace is dense

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

Related facts

 * Open subset of open subspace is open
 * Closed subset of closed subspace is closed