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A trivial, yet difficult question. How would one prove that the real numbers are not compact, only using the definition of being compact? In other words, what happens if we reduce an open cover of R to a finite cover of R?

I let V be a collection of open subset that cover R

Then I make the assumption that that this open cover can be reduced to a finite subcover.

(Clearly this is not possible) I am struggling to see/show what happens to R when I make this assumption. Should I simply find a counterexample, if so, what could it look like?

I have proven several, similar problems, but this one is so general that its tricky.

I let V be a collection of open subset that cover R

Then I make the assumption that that this open cover can be reduced to a finite subcover.

(Clearly this is not possible) I am struggling to see/show what happens to R when I make this assumption. Should I simply find a counterexample, if so, what could it look like?

I have proven several, similar problems, but this one is so general that its tricky.

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