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I'm self-studying Rudin's Principles of Mathematical Analysis, and making an attempt to prove the theorems in the book myself before reading how they're done. I'm stuck on Theorem 2.43:

Let $P$ be a nonempty perfect set in $\mathbb{R}^{k}$. Then $P$ is uncountable.

I think I understand what perfect means (that a set is made entirely up of and contains all its limit points), and I understand that I need to show that no bijection exists between the set of elements of $P$ and the positive integers. I considered trying to find a surjection from the set of points in $P$ to the set of all integer sequences or the set of all sequences made up of 0 and 1, as well as from $P$ to some subset of the real number line, but I can't figure out how to go about actually proving the result I need. Maybe there's a connection to compact sets I'm missing? Any small hints would be appreciated (I still am hoping to have some of the fun of finding an answer).

i like math
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    if you have the book, why not peek at it to gain some inspiration? – peek-a-boo May 04 '21 at 05:31
  • @peek-a-boo I would, but often the main idea of the proof is given right at the beginning, and I'd like to refrain from spoiling the whole thing if I possibly can. – i like math May 04 '21 at 05:33
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    suppose it's countable and try to derive a contradiction (usually when trying to prove something is uncountable, it's easier to assume it is countable and get a contradiction; that's for example how we prove (atleast the proofs I know) that the real numbers are uncountable, the cantor set is uncountable etc). If you're stuck at any point ask yourself "have I used the property of perfectness"? How does compactness help? – peek-a-boo May 04 '21 at 05:37
  • Okay, I'll try focusing my efforts on coming up with a contradiction. Thanks – i like math May 04 '21 at 05:48

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Suppose $P$ is countable, so that $P = \{p_1,p_2, p_3, \ldots\}$ is a complete enumeration of $P$ and then show there must be a point in $P$ that is unequal to any $p_n$ by its construction, to achieve a contradiction. You can apply a recursive construction to achieve it. Use completeness or local compactness of $\Bbb R^n$ to show the existence of such a point.

Henno Brandsma
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