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i am not sure which one = real analysis in the US. but about the adcoms:
of the 3 courses you mention the first one sounds like "introduction to proofs", not bad, but not that impressive for an adcom. the second one "principles" is kind of ambiguous, but also if i were sitting on an adcom it would sound like an introductory course at best about calculus. btw, from the syllabus it sounds like thats exactly what it is.
real analysis is the only course that sounds like its on a very high level. of course, there are real analysis courses on a lower level, but i think this is where your schools reputation plays a role in signaling to the adcoms what kind of course this would be.
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