The following two lemmas provide motivation towards studying the real numbers: $\mathbb{Q}$ has gaps, so physical phenomena is not always be realizable in $\mathbb{Q}$.
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Lemma. There is no $p \in \mathbb{Q}$ such that $p^2 = 2$.
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Lemma. The set $A = \{p \in \mathbb{Q} \space | \space p^2 < 2 \}$ has no largest element and the set $B = \{p \in \mathbb{Q} \space | \space p^2 > 2\}$ has no least element.
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The second lemma states that $\sqrt{2}$ can be arbitrarily approximated in $\mathbb{Q}$, but never reached; i.e. that there is a gap between $A$ and $B$.
An ordered set $S$ is a set equipped with a relation $<$, called a total order, which has the properties:
$$ \textbf{Law of Trichotomy. } \text{Exactly one is true: $x<y , y<x, x=y$} \\ \textbf{Transitivity. } x<y \wedge y <z \implies x < z \hspace{23pt} $$
For any subset $E$ of an ordered set $S$, the boundedness of $E$ is characterized below.
$$ \exists \beta \in S : \forall x \in E , x \leq \beta \iff \text{$E$ is bounded above \& $\beta$ is an upper bound of $E$} $$
The notion of a lower bound and lower-boundedness can be defined analogously.
For any subset $E \sube S$ which is bounded above, the notion of a least upper bound is defined as the supremum of $E$.
$$ \beta = \sup E \text{ if and only if } \textbf{both }\text{of the following conditions hold:} \\ \begin{cases} \textbf{1.} \text{ $\beta$ is an upper bound of $E$.} \\ \textbf{2.} \text{ If $\alpha \in S$ s.t. $\alpha < \beta$, then $\alpha$ is not an upper bound of $E$.}\end{cases} $$
The notion of a greatest lower bound can be defined analogously and is known as the infimum of $E$, denoted $\inf E$.
An ordered set $S$ is said to have the least upper bound property if all of its nonempty, bounded above subsets have supremums in $S$.
$$ \text{For any bounded above subset $\empty \neq E \sube S, \space \sup E \in S$.} $$