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I'm surprised no one has said Marsden and Hoffman's "Elementary Classical Analysis", but perhaps it is too elementary or classical. I didn't learn from it as an undergrad, but I did find myself turning to it as I worked problems from "Berkeley's Problems in Mathematics" by de Souza and Silva. M&H fleshes out a lot more detail, which Rudin spares for the sake of elegance or relegates to the exercises. I wish I had it or Korner's book "A First Second or Second First Course in Analysis" alongside Rudin when I first studied analysis. In particular, I prefer Marsden and Hoffman's treatment of Arzela-Ascoli over Rudin's.