# Real analysis

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**Real analysis** is an area of mathematics that deals with sets and sequences of real numbers, as well as functions of one or more real variables. As one of the main branches of analysis, it can be seen as a subset of complex analysis, many results of the former being special cases of results in the latter.

As an academic subject, real analysis is typically taken in college after a two- or three-semester course in calculus, and usually after a course in rigorous mathematical proof. As such, it can be seen as a generalization of calculus to higher dimensions and to more general types of functions.

Some major concepts in real analysis include:

- fundamental calculus notions such as limits, continuity, derivatives, integrals, and the convergence and divergence of infinite series
- sequences of sets and unions and intersections of arbitrary numbers of sets
- least upper bound and greatest lower bound of a set
- elementary notions of topology, including open, closed, countable, connected, and compact sets
- liminf and limsup, respectively the "limit inferior" and "limit superior" of a sequence
- Cauchy sequences and their relation to convergent sequences
- metrics and metric spaces, which generalize the notions of distance and Euclidean spaces
- pointwise convergence and uniform convergence of sequences of functions
- rates of convergence and "Big O notation"
- sigma algebras, measures and measure spaces

Some important results in real analysis include:

- basic calculus results such as the fundamental theorem of calculus, intermediate value theorem, mean value theorem, and monotone convergence theorem
- Bolzano-Weierstrass theorem
- Heine-Borel theorem
- inverse function theorem and implicit function theorem
- Fubini's theorem
- Banach fixed point theorem
- and various inequalities: