# Real Analysis

Universities are offering this course as one course or some time they are offering it in parts.

### Course objects

At the end of this course the students will be able to uunderstand the basic set theoretic statements and emphasize the proofs’ development of various statements by induction. Define the limit of, a function at a value, a sequence and the Cauchy criterion. Prove various theorems about limits of sequences and functions and emphasize the proofs’ development. Define continuity of a function and uniform continuity of a function, prove various theorems about continuous functions and emphasize the proofs’ development. Define the derivative of a function, prove various theorems about the derivatives of functions and emphasize the proofs’ development. Define a cluster point and an accumulation point, prove the Bolzano-Weierstrass theorem, Rolles’s Theorem, extreme value theorem, and the Mean Value theorem and emphasize the proofs’ development. Define Riemann integral and Riemann sums, prove various theorems about Riemann sums and Riemann integrals and emphasize the proofs’ development.

### Course contents

The Real Number System: Ordered Fields. The Field of Reals. The Extended Real Number System. Euclidean Space. Numerical Sequences and Series. Limit of a Sequence. Bounded Sequences. Monotone Sequences. Limits Superior and Inferior. Subsequences. Limit of a Function and Continuous Functions. Uniform Continuity. Kinds of Discontinuities. Derivable and Differentiable Functions. Mean Value theorems. The Continuity of Derivatives. Taylor's theorem. Riemann Stieltijes Integrals: Definition, Existence and Properties of the Riemann Integrals. Integral and Differentiation.