Isometry Preserves Sequence Convergence

Theorem
Let $M_1 = \left({S_1, d_1}\right)$ and $M_2 = \left({S_2, d_2}\right)$ be metric spaces or pseudometric spaces.

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

Let $\left\langle{x_n}\right\rangle$ be an infinite sequence in $S_1$.

Suppose that $\left\langle{x_n}\right\rangle$ converges to a point $p \in S_1$.

Then $\left\langle{\phi \left({x_n}\right)}\right\rangle$ converges to $\phi \left({p}\right)$.