Best First Course in Real Analysis Book for Self Study

Question

Does anybody know of a good book in real analysis for self study for a beginner? What about Analysis 1 by Terence Tao?

Answer 1

Having had my first course in real analysis taught from Tao's Analysis I, I can honestly say that, for a beginner, Tao's book is a great resource. Tao start from the absolute beginning, setting up the Peano postulates and then constructing $\mathbb{N},\mathbb{Z},\mathbb{Q}$ and $\mathbb{R}$ one by one. From the Peano postulates one has the natural numbers $\mathbb{N}$, the increment operation and mathematical induction. Addition is then defined in terms of repeated incrementation and the various properties of it are derived. $\mathbb{Z}$ is developed as the completion of $\mathbb{N}$ with respect to equations like $5+a=3$ and $n+b=0$, where $n \in \mathbb{N}$, and it's properties are proved, either in the text or as an exercise. $\mathbb{Q}$ is constructed from $\mathbb{Z}$ as it's completion with respect to division and yet again its properties are developed. Tao then introduces sequences and Cauchy sequences and constructs the real numbers as a completion of $\mathbb{Q}$, though this is a bit more conceptual.

Thus, Tao assumes no real previous knowledge of these number systems and their properties. Tao also develops all the necessary set theory along the way. This really helped me later to understand sequences, series, derivates and integrals and their properties, as I had a good feeling for the number systems we were working over because Tao had built all the machinery up step by step.

Overall, a great introduction.