PROOFS. Welcome to Mathematics

Welcome to Mathematics!The book is meant to serve as an introduction to higher mathematics for future mathematicians and future teachers of mathematics. For a lot of the readers, this will be a prequel for a lot of mathematics they have learned before.Proof-writing. Proof-discovering.The basic topics such as Euclidean geometry and modular arithmetic.The axiomatic approach. To illustrate it, we provide a few samples of ``abstract'' mathematics: equivalence relations, products, quotients, groups, metric spaces, order spaces.Instead of ``a wall of text'', the style is that of a classroom presentation. The presentation is also modular. The book is self-contained. This makes it a good choice for both teaching and self-study.The new ideas are initially illustrated with examples from pre-calculus. The examples from calculus are only secondary and can be skipped.We solve the easy problems the hard way. Later, the hard way with the hard problems is the easy way. Read more