Isometry Preserves Sequence Convergence

Theorem
Let $M_1 = \struct {S_1, d_1}$ and $M_2 = \struct {S_2, d_2}$ both be metric spaces or pseudometric spaces.

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

Let $\sequence {x_n}$ be an infinite sequence in $S_1$.

Suppose that $\sequence {x_n}$ converges to a point $p \in S_1$.

Then $\sequence {\map \phi {x_n}}$ converges to $\map \phi p$.