The real analysis lifesaver : all the tools you need to understand proofs /Raffi Grinberg

Preliminaries. Introduction ; Basic Math and Logic ; Set Theory --
Real Numbers. Least Upper Bounds ; The Real Field ; Complex Numbers and Euclidean Spaces --
Topology. Bijections ; Countability ; Topological Definitions ; Closed and Open Sets ; Compact Sets ; The Heine-Borel Theorem ; Perfect and Connected Sets --
Sequences. Convergence ; Limits and Subsequences ; Cauchy and Monotonic Sequences ; Subsequential Limits ; Special Sequences ; Series.

Real analysis is difficult. For most students, in addition to learning new material about real numbers, topology, and sequences, they are also learning to read and write rigorous proofs for the first time. The Real Analysis Lifesaver is an innovative guide that helps students through their first real analysis course while giving them the solid foundation they need for further study in proof-based math. Rather than presenting polished proofs with no explanation of how they were devised, The Real Analysis Lifesaver takes a two-step approach, first showing students how to work backwards to solve the crux of the problem, then showing them how to write it up formally. It takes the time to provide plenty of examples as well as guided "fill in the blanks" exercises to solidify understanding. Newcomers to real analysis can feel like they are drowning in new symbols, concepts, and an entirely new way of thinking about math. Inspired by the popular Calculus Lifesaver, this book is refreshingly straightforward and full of clear explanations, pictures, and humor. It is the lifesaver that every drowning student needs.