Functions and Analysis

    

Correcting Rudin's Perfect Sets Are Uncountable Proof

Authors: Timothy W. Jones

It is fascinating fact that the reals are uncountably infinite. Usually Cantor's diagonal method is used to show this. Rudin gives a second proof that promises to be more rigorous than this method. But his proof is a little confusing, if not incorrect. His proof does not stipulate that the perfect set be bounded, but its proof hinges on a local, bounded phenomenon. We duplicate Rudin's proof and argue using two examples that assuming any indexing scheme for the presumed countable set can't work. We then give two proofs: one re-indexes points and the other indexes in the course of the proof.

Comments: 4 Pages.

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Submission history

[v1] 2023-08-13 14:11:14

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