Metrizable implies hereditarily normal

Statement
Any metrizable space (i.e., any topological space that arises from a metric space) is a hereditarily normal space -- every subspace of it is a normal space.

Facts used

 * 1) uses::Metrizable implies normal
 * 2) uses::Metrizability is hereditary
 * 3) uses::Metrizable implies monotonically normal
 * 4) uses::Monotonically normal implies hereditarily normal
 * 5) uses::Metrizable implies perfectly normal
 * 6) uses::Perfectly normal implies hereditarily normal

Proof from facts (1) and (2)
By fact (2), every subspace of a metrizable space is metrizable, and thus, by fact (1), every subspace of a metrizable space is normal.

Proof by facts (3) and (4)
By fact (3), any metrizable space is monotonically normal, hence by fact (4), it is hereditarily normal.

Proof by facts (5) and (6)
By fact (5), any metrizable space is perfectly normal, hence by fact (6), it is hereditarily normal.