# Real Analysis

## Real Sequences

Sequence of real numbers A sequence of real numbers (or a real sequence) is defined as a function $f: \mathbb{N} \to \mathbb{R}$ , where $\mathbb{N}$ is the set of natural numbers and $\mathbb{R}$ is the set of real numbers. Thus, $f(n)=r_n, \ n \in \mathbb{N}, \ r_n \in \mathbb{R}$ is a

## Smart Fallacies: i=1, 1= 2 and 1= 3

This mathematical fallacy is due to a simple assumption, that $-1=\dfrac{-1}{1}=\dfrac{1}{-1}$ . Proceeding with $\dfrac{-1}{1}=\dfrac{1}{-1}$ and taking square-roots of both sides, we get: $\dfrac{\sqrt{-1}}{\sqrt{1}}=\dfrac{\sqrt{1}}{\sqrt{-1}}$ Now, as the Euler’s constant $i= \sqrt{-1}$ and $\sqrt{1}=1$ , we can have $\dfrac{i}{1}=\dfrac{1}{i} \ldots \{1 \}$ $\Rightarrow i^2=1 \ldots \{2 \}$ . This

## Set Theory, Functions and Real Numbers

These study notes on Set Theory, Functions and Real Numbers were written by Gaurav Tiwari when he was studying as a Math undergraduate in 2012-2013. The language is sought to be simple and easy to understand. Further reading material is also provided with this article. If you have any questions, feel free to send a

## Getting Started with Measure Theory

Last year, I managed to successfully finish Metric Spaces, Basic Topology and other Analysis topics. Starting from the next semester I’ll be learning more pure mathematical topics, like Functional Analysis, Combinatorics and more. The plan is to lead myself to Combinatorics by majoring Functional Analysis and Topology. But before all those, I’ll be studying measure theory and

## Irrational Numbers and The Proofs of their Irrationality

“Irrational numbers are those real numbers which are not rational numbers!” Def.1: Rational Number A rational number is a real number which can be expressed in the form of where $a$ and $b$ are both integers relatively prime to each other and $b$ being non-zero. Following two statements are equivalent to the

## The Area of a Disk

If you are aware of elementary facts of geometry, then you might know that the area of a disk with radius $R$ is $\pi R^2$ . The radius is actually the measure(length) of a line joining the center of disk and any point on the circumference of the disk or any other circular

## Triangle Inequality

Triangle inequality has its name on a geometrical fact that the length of one side of a triangle can never be greater than the sum of the lengths of other two sides of the triangle. If $a$ , $b$ and $c$ be the three sides of a triangle, then neither $a$