Date: Jul 9, 2013 5:53 PM
Author: Paul
Subject: Problem understanding proof of Jordan Curve Theorem

I have been reading Tverberg's proof of the Jordan Curve Theorem: 
http://www.maths.ed.ac.uk/~aar/jordan/tverberg.pdf

I understand completely the proofs of lemmas 1 and 2, but I'm having trouble following the proof of lemma 3. In particular, it's not clear to me why there is a disc D with abs(a - b) maximal. This is probably a consequence of the fact that continuous functions on compact sets attain their maxima, but I can't justify that assertion.

Thank you very much for your help,

Paul Epstein