That textbook was the assigned reading in my proofs class and I thought it was one of the worst textbooks I've ever seen, both as a first exposure to proofs and as an analysis text. To OP, if you feel the same, I'd suggest How to Prove It by Velleman for a proofs book, (Amazon Link). For analysis, I would recommend Terence Tao's Analysis I and Analysis II, Abbott's Understanding Analysis, or even Baby Rudin over that book.
That is what I thought you'd say.
...teach a bitizen to fish and you feed him for life.
+1.
OP: You might also want to brush up on your proof skills, in case they're rusty. How to Prove It: A Structured Approach by Velleman is a good place to start.