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Let's suppose we have this version of BW theorem:

Theorem: Every infinite bounded subset of $\mathbb{R}$ has an accumulation point.

Is there any way, using this (Or the every bounded sequence has a convergent subsequence version) theorem, to prove that every uncountable subset of $\mathbb{R}$ has an accumulation point?

Does this generalize to $\mathbb{R}^n$?

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Hint: Let $E\subset \mathbb R$ be uncountable. Then $E=\cup_{n=1}^\infty(E\cap [-n,n]).$

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  • $\begingroup$ Since $E$ is uncountable, and a countable union of some sets $E\cap [-n,n]$, at least one of these sets is uncountable, and by construction bounded. Then I can apply BW. Am I right? Thanks. $\endgroup$ – David Molano Mar 1 '16 at 23:28
  • $\begingroup$ Yes that's right. $\endgroup$ – zhw. Mar 1 '16 at 23:38

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