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I know this proof is somewhat similar, or related to the Baire's Category Theorem but I can't seem to figure out how to do it.

The Baire Category theorem asserts that if X is a complete metric space or a locally compact Hausdorff space (my case), then the complement of a countable union of nowhere dense sets is always nonempty.

How can I use this on the proof?

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Since every compact Hausdorff space is locally compact, you can actually use the statement that you mentioned.

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