Isometry Preserves Sequence Convergence

Theorem
Let $M_1=(S_1,d)$ and $M_2 = (S_2,d)$ be metric spaces or pseudometric spaces.

Let $\phi\colon S_1 \to S_2$ be an isometry.

Let $\langle x_n \rangle$ be an infinite sequence in $S_1$.

Suppose that $\langle x_n \rangle$ converges to a point $p \in S_1$.

Then $\langle \phi(x_n) \rangle$ converges to $\phi(p)$.