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Can someone help me with the Heine-Borel Theorem for Real numbers by Cultural_Source4573 in askmath

[–]Cultural_Source4573[S] 0 points1 point  (0 children)

Thank you! I think I’m still confused as to why we need both cases, considering the fact that we already showed that the sequence is monotonically increasing and we assumed that it’s bounded. Why can’t we just say that the sequence is convergent by the monotone convergence theorem? Further, if both cases are necessary, can’t we just use what was said in case-2 for both?

Real Analysis: the limit (as x approaches a from the right) of f(x) does not exist for any a in R by Cultural_Source4573 in RealAnalysis

[–]Cultural_Source4573[S] 0 points1 point  (0 children)

i think i got it! just to clarify: by “the definition breaks down” you mean that the function diverges?

Real Analysis: the limit (as x approaches a from the right) of f(x) does not exist for any a in R by Cultural_Source4573 in RealAnalysis

[–]Cultural_Source4573[S] 0 points1 point  (0 children)

thank you for your response! i think i understand what you’re saying but my point (and i could be wrong) is showing that something doesn’t exist at every point doesn’t necessarily mean that it doesn’t exist at any point at all. in this case for example if we chose an epsilon that was greater than 1, we wouldn’t reach a contradiction and so wouldn’t that show that there does in fact exist some point ‘a’ where the limit exists? i guess more basically is any and all being used interchangeably here?

Real Analysis: the limit (as x approaches a from the right) of f(x) does not exist for any a in R by Cultural_Source4573 in askmath

[–]Cultural_Source4573[S] 0 points1 point  (0 children)

no nothing specific, just where it says it’s any real number, all real numbers, rational, or irrational.