Isometry Preserves Sequence Convergence

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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 $\map \phi p$.

Proof 1

\(\ds \)

\(\)

\(\ds \lim_{n \mathop \to \infty} \map {d_2} {\map \phi {x_n}, \map \phi p}\)

\(\ds \)

\(=\)

\(\ds \lim_{n \mathop \to \infty} \map {d_1} {x_n, p}\)

Definition 1 of Isometry (Metric Spaces)

\(\ds \)

\(=\)

\(\ds 0\)

Definition 3 of Convergent Sequence

Hence, by definition, $\sequence {\map \phi {x_n}}$ converges to $\map \phi p$.

$\blacksquare$

Proof 2

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