Question

How do I find the greatest lower bound of a set?

Answer 1

A set is bounded if it is bounded both from above and from below. A set that is bounded from below has lower bounds. Every lower bound is less than or equal to the greatest lower bound.

Complete step by step solution:
A set is said to be bounded from below if it has lower bounds. Every element in the set is greater than or equal to the lower bounds. The greatest lower bound of a set is a lower bound of the set that is greater than or equal to the lower bounds of the set.
Consider a non-empty subset $\text{S}$ of the set of real numbers $\mathbb{R}.$
The set $\text{S}$ is said to be bounded from below if there exists a number $w\in \mathbb{R}$ such that $s\ge w$ for all $s\in \text{S}\text{.}$ Each number $w$ is called a lower bound of $\text{S}\text{.}$
If the set $\text{S}$ is bounded below, then the number $w$ is said to be the greatest lower bound of $\text{S}$ if it satisfies the conditions:
$\left( \text{i} \right) w$ is a lower bound of $\text{S}$
and $\left( \text{ii} \right)$ if $t$ is any lower bound of $\text{S,}$ then $t\le w.$
In order to find the greatest lower bound of a set, we need to find the lower bounds of the set. Because the greatest lower bound itself is a lower bound which is greater than or equal to every other lower bounds of the set.

Note: Let us recall the following lemma.
Lemma $1.$ A number $w$ is the greatest lower bound of a non-empty subset $\text{S}$ of $\mathbb{R}$ if and only if $w$ satisfies the conditions:
$\left( \text{i} \right) s\ge w \forall s\in \text{S,}$
$\left( \text{ii} \right) if \, l>w, \, then\, \exists \,t \, \in\, \text{S} \,such\, that\, l>t.$
The greatest lower bound is also called the infimum.