Introduction to real analysis (1.0) - Preliminaries

(I am wary of the fact that I have not been posting anywhere near $\frac{3}{2}$ times per day, hopefully over the next week I will be able to pull my average up somewhere closer to $\frac{3}{2}$)

This post will contain several definitions/useful concepts that will be frequently used in the series of posts titled 'Introduction to real analysis'.

Definition 1 (Limit of a function) - A real valued function $f$ defined on some domain $D \subset \mathbb {R}$ is said to have a 'limit as $x$ approaches $a$' for some $a \in D$ if the following statement is true:

' There exists a real number $L>0$ such that the following statement is true:

 For all $\epsilon > 0 $ there exists a $\delta > 0 $ such that if $0 < |x - a| < \delta$, then $|f(x) - L| < \epsilon $

Intuitively, this means that given we take points close enough to $a \in D$, $f(x)$ can be made arbitrarily close to $L$.

Definition 2 (Continuity of a function) - A real valued function $f$ defined on some domain $D \subset \mathbb {R}$ is said to be 'continuous at $a \in D$' if the following statement is true:

 For all $\epsilon > 0 $ there exists a $\delta > 0 $ such that if $ |x - a| < \delta$, then $|f(x) - f(a)| < \epsilon $

Definition 3 (Continuity on a set) - A real valued function $f$ with domain $D \subset \mathbb {R}$ is said to be continuous on a set $S \subset D$ if for all $x \in S$, $f$ is continuous at $x$.

Visually, a function that is continuous contains no 'holes' and no 'jumps'. Most of the popular functions, $sin(x)$, $cos(x)$, $a^x$ are continuous in $\mathbb {R}$.

Definition 4 ( Norm in $\mathbb {R}$) - The 'distance' or 'absolute' value function is defined as follows:

$|x| = x$ if $x \geq 0$, $|x| = -x$ if $x < 0$.

Visually, $x$ is the 'distance of $x$ from $0$'.

Theorem 1 (Triangle Inequality) - For any $a,b,c \in \mathbb {R}$ the following inequality holds:

$|a+b| \leq |a| + |b|$

A proof of Theorem 1 will not be provided at the moment, (perhaps it will be provided in a later post). A sketch however, of one possible proof, has been provided below:

Proof Sketch (Theorem 1) - Consider the cases when $a \geq 0$, $a<0$, $b \geq 0$, $b<0$

Prove Theorem 1 for all combinations of aforementioned cases.

Theorem 2 (Reverse Triangle Inequality) - For any $a,b,c \in \mathbb {R}$ the following inequality holds:

$| |a| - |b| | \leq |a + b|$.

One can similarly prove Theorem 2.


THIS 'PRELIMINARY' LIST IS NOT CONCRETE, AND MAY BE SUBJECT TO UPDATE.