Introduction
You have already learned some concepts of calculus such as limit of a sequence, limit, continuity, derivative, and integral of a function etc. Real Analysis studies them more rigorously using a language based on axioms and definitions (remember $\epsilon-\delta$ definition of limit?). It may be challenging to many students to get used to the language and axiomatic approach. But being comfortable with the machinery of Real Analysis would definitely make you a better mathematician.

Pure mathematics is, in its way, the poetry of logical ideas. One seeks the most general ideas of operation which will bring together in simple, logical and unified form the largest possible circle of formal relationships. In this effort toward logical beauty spiritual formulas are discovered necessary for the deeper penetration into the laws of nature. -- Albert Einstein, Obituary for Emmy Noether (May 5, 1935).

Set notation:

• \(\varnothing\) or \(\{\}\), the empty set or null set.


• \(\mathbb N \text{ or } \mathbb Z^+=\{1,2,3,\ldots\}\), the set of natural numbers.


• \(\mathbb Z=\{0,\pm1,\pm2,\pm3,\ldots\}\), the set of all integers.


• \(\mathbb Q=\{\frac{p}{q}\;|\;p,q\in \mathbb Z\text{ and }q\neq 0\}\), the set of all rational numbers.


• \(\mathbb R\), the set of all real numbers.


• \(\mathbb C\), the set of all complex numbers.


• $\displaystyle\sum_{i=1}^na_i=a_1+a_2+\ldots+a_n$. E.g., $\displaystyle\sum_{i=1}^n i=\frac{n(n+1)}{2}$
• $\displaystyle\prod_{i=1}^na_i=a_1a_2\cdots a_n$. E.g., $\displaystyle\prod_{i=1}^n i=:n!$

Set operation notation:

• $A\subseteq B$ : read as "$A$ is a subset of $B$".


• $A\subset B$ : read as "$A$ is a proper subset of $B$".


• $A\cup B=\{x\;|\; x\in A \text{ or } x\in B\}$.


• $A\cap B=\{x\;|\; x\in A \text{ and } x\in B\}$.


• $A\setminus B=\{x\;|\; x\in A \text{ and } x\notin B\}$.

Frequently used notation in Real Analysis:

• $x\in S$ : read as "$x$ belongs to $S$" or "$x$ is in $S$".


• $\forall\; x\in S$ : read as "for all $x$ in $S$".


• $\exists\; x\in S$ : read as "there exists $x$ in $S$".


• $\operatorname{\;s.t.\;}$ : read as "such that".

Last edited by Dr. Mallik on