IMO, Rudin is difficult not because of its proofs or lack of them (many proofs in discrete math can be no less brutal than anything in Rudin), rather that it's almost completely and utterly devoid of illuminating examples. For example, the definitions of "neighborhood", "limit point", "closed set", "open set", "bounded set", "perfect set", dense set" are crammed into a single definition 2.18 in chapter 2(Topology in Euclidean Spaces) in 3rd edition. The rest of the chapter is made up of theorems and corollaries. No related examples. On the other hand, Raffi Grinberg's analysis book meant to guide one through Rudin's book spends a whole chapter on elaborating on 2.18. And to be honest even that is barely adequate (totally inadequate, actually) if one wishes to become technically proficient in dealing with basic concepts in analysis with ease (that requires exposure to lots and lots of different examples). Although, probably, neither book has the latter as their goal.