Analysis I or Real Analysis serves as foundation of our rigorous mathematical study, where results seem obvious and fairly intuitive and that's not the case when it comes to prove those results. In this course, we learn how to use axioms, definitions and theorems effectively to prove desired results.
For example, consider something as intuitively fundamental as the associative property of addition applied to the series $\sum_{n = 1}^{\infty} (-1)^n$. Grouping $$(-1 + 1) + (-1 + 1) + \cdots = 0 + 0 + \cdots = 0,$$ whereas grouping in another yields $$\begin{aligned} -1 + (1 - 1) + (1 - 1) + \cdots &= -1 + 0 + 0 + \cdots \\ &= -1. \end{aligned}$$
Manipultations that are legitimate in finite settings do not always extend to infinite settings. Deciding when they do and when they do not is one of the central themes of analysis.
I started this course with Prof. Abbott's book Understanding Analysis in hand and while studying I came across Prof. Baisheng Yan's well-crafted notes based on Understanding Analysis for MTH 320 (Spring 2018). Unfortunately, Spring 2018 version of the course isn't available online. I managed to find Spring 2019 version that was taught by Prof. Matthew Cha and there are many homework assignments, their solutions and exams available for free.
I'm using these resources to learn Real Analysis. This page serves as a navigation tool to access materials quickly. I will be working and writing proofs of various exercises given in both book and assignments to get most out of the course and to learn proof-writing skills. If you find any mistake in a proof or much nicer proof of some problem, then feel free to reach out or make a pull request.
All personal chapterwise write-ups can be found in Github Repository of this site.
These notes use Obsidian's Markdown/Math rendering. GitHub's rendering may not render inline LaTeX or certain mathematical notation (e.g., braces) exactly as in Obsidian. If something looks off, open the files in Obsidian for the intended formatting.