Two results I have seen used without proof in undergraduate lectures were:
Tychonoff's theorem in I don't remember what courses (but can be many).
IIRC Carathéodory's extension theorem in probability theory (but might also have been a different fact from measure theory).
In both cases the proofs are not long, but were deemed not useful enough to be taught. I am wondering whether this is special to the courses I had or generally common.
Also, various courses on graph theory use some versions of the Jordan curve theorem; even the ones not requiring analysis (speaking of piecewise linear paths) are usually not proven. And several analysis courses don't prove the basic properties of real numbers, instead treating them as axioms.
