Isometry Preserves Sequence Convergence/Proof 2
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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
This theorem requires a proof. In particular: proof from the fact that an isometry is a homeomorphism You can help $\mathsf{Pr} \infty \mathsf{fWiki}$ by crafting such a proof. To discuss this page in more detail, feel free to use the talk page. When this work has been completed, you may remove this instance of {{ProofWanted}} from the code.If you would welcome a second opinion as to whether your work is correct, add a call to {{Proofread}} the page. |