so far in my mathematical education the proofs i've had trouble with the most have been the proof to Urysohn's lemma (the idea behind it is simple, but the execution took some time to process) and Arcela-Ascoli's theorem, with some functional analytic stuff trailing after (open mapping theorem, Mazur separation theorem). our Analysis 3 class had a bunch of difficult ones that i haven't mastered to this day, like Caratheodory's extension theorem (which i can outline in broad strokes, but still don't get perfectly), the proof of multivariable "change of variables", Gauss' divergence theorem (this one was easier, but i have yet to process it in its entirety) and Fubini's theorem (which i think i've mastered recently)