My university had a proof based, let's construct the natural numbers from axioms and prove that a+b == b+a type course as the first class for aspiring math majors. It was a "weed out" course, not in the sense that it was particularly hard, but more in the sense of introducing students to this is what 'real' math is actually about, and not so much the stuff you were doing in high school. Some students decided each year that they didn't want to do 'real' math and went on to focus on other things.

Yep.

At my university, after the usual calculus/diff eq/lin alg sequence everyone in STEM takes, we had “intro to advanced mathematics” that was proof based, taught in the spring, and a pre-req for everything higher level (abstract algebra, real analysis, etc). Most math programs have a similar “first proofs” class as their “weed out” class.

Math isn't popular enough to have weed out courses, and math professors of higher division classes don't mind failing half the class. There isn't much of a drawback to letting as many people who want to major in math.

But intro STEM math is used as a weed out for other majors. You aren't going to get far in CS if you aren't able to ace your basic calculus classes.

Real analysis or other similar "proofs of basic things" classes usually fill this role.