Is there any application of the Heine-Borel theorem or the Bolzano-Weirstrass theorem to Economics? Also where are the notions of compact sets and elementary Real Analysis used in Economics?
There are too many applications of the concepts and results you mention for a list to be of any use. As a first introduction, look at:
There used to be pdfs of (reasonably high quality) working version of the book available online, but they are harder to find now that the book is out (I believe the author put these working versions online as a way to get feedback and typo-reports but removed them since then). If you look hard enough, some of these working version might still live in corners of the internet though.
Also have a look at this related question: What are some applications of Real Analysis in Graduate Economics?