Problem Solving Practice

Charles W. Trigg — Mathematical Quickies: $270$ Stimulating Problems with Solutions.

At the moment I read: Daniel J. Velleman, How to Prove It: A Structured Approach.

The first 2 chapters contain propositonal logic, quantfiers and basic set operations. From then on it's all about proofs: In chapter 3 the author explains various proof techniques step by step and in chapters 4,5,6,7 on relations, functions, induction and infinite sets you get to use these. At the end of each subchapter (I'm currently at 3.7) there are 10-30 exercises in varying difficulty - till now most are of exactly the right difficulty.

I'm using it to prepare myself for university and until now I'm very happy with it, although I don't know how much the difficulty increases in the later chapters.

If you want a solid introduction to rigorous, proof based math it seems to be not too bad of a choice, your problem solving skills will definitely increase (especially if you work through the hundreds of exercises).