The oddity is that I could read proofs for both subjects: the reasoning made sense. But I couldn't develop a proof.
What was yours like? The one I took didn't cover much on the actual writing of proofs. The professor accepted reasonable essays with high-school level notation. Instead, he gave us a toolbox for proving things: pairing terms in a series to find the sum, rewriting recursive equations, etc. It was mostly to show that clever tricks are how mathematicians prove new things. But that might've been because the professor was a guy who reveled in clever solutions.
What I personally recall was following along in class while the professor walked through classic proofs emphasizing what each new notation meant as well as the difference between Direct, indirect, and induction proofs. Tests and assignments were essentially recreating proofs cherry picked to be similar to ones we walked through in class.
I realize that's kind of how all my higher level math classes were run. It's just that once the cherry picking becomes looser, intuition doesn't necessarily catch up :(.
I'd love to know how to develop an intuition for writing proofs from scratch.
Basically you sort of write the broad strokes of the proof up front, leaving the right hand sides of statements like "Choose epsilon such that epsilon = __" blank. Then you do a bunch of scratch work to figure out what epsilon needs to be so that your proof works out in the end.
Another challenge with analysis is that inequalities are central, so fluency in their manipulation is absolutely critical. And naturally most students aren't fluent with them by the time they take analysis, so they get bulldozed by Baby Rudin and learn to hate a pretty cool (and useful) branch of math