3.3 Subsets of the Real Line

It is helpful to note at this point that the completeness property from before also guarantees the existence of infimums for non-empty subsets that are bounded below.

Theorem 3.3

Every non-empty subset of $\mathbb{R}$ that is bounded below has an infimum in $\mathbb{R}$ .

Proof.

Let $S$ be a non-empty subset of $\mathbb{R}$ that is bounded above. Then by the completeness property there exists $\ell\in\mathbb{R}$ such that $\ell=\sup(S)$ . Now consider the set $S^{\prime}=\{-s\,:\,s\in S\}$ . It is clear that $S^{\prime}$ is a non-empty subset of $\mathbb{R}$ which is bounded below, and that $-\ell=\inf(S)$ . ∎

3.3.1 Intervals

Quite often we will need some compact notation for denoting a continuous subset of $\mathbb{R}$ , such as the positive real numbers, all real numbers between -1 and 4, etc. These subsets are called intervals, and we will define them now.

Definition 3.2.

Let $a,b\in\mathbb{R}$ with $a<b$ . Then the closed interval from $a$ to $b$ is given by

[a,b]=\{x\in\mathbb{R}:a\leq x\leq b\},

and the open interval from $a$ to $b$ is given by

(a,b)=\{x\in\mathbb{R}:a<x<b\}.

Likewise the half-open intervals may be defined similarly as

(a,b]=\{x\in\mathbb{R}:a<x\leq b\},

[a,b)=\{x\in\mathbb{R}:a\leq x<b\}.

To denote a half-unbounded or fully unbounded interval, we can use a $\infty$ symbol. For example:

[a,\infty)=\{x\in\mathbb{R}:a\leq x\},

(-\infty,0)=\{x\in\mathbb{R}:x<0\},

and we could also say

(-\infty,\infty)=\mathbb{R}.

It is also useful to note that

\sup((a,b))=\sup([a,b])=b,\quad\inf((a,b))=\inf([a,b])=a