Sequence Converges to Point Relative to Metric iff it Converges Relative to Induced Topology

Theorem
Let $M=(S,d)$ be a Metric Space or a Pseudometric Space.

Let $T=(S,\tau)$ be the induced topological space.

Let $\langle x_n \rangle$ be an infinite sequence in $S$.

Let $p \in S$.

Then $\langle x_n \rangle$ converges to $p$ relative to $d$ iff it does so relative to $\tau$.

Proof
Suppose $\langle x_n \rangle$ converges to $p$ relative to $d$.