\section{Applications of the Supremum Property}

\begin{theorem}[\textbf{Archimedian Property}]
	If $x \in \R$, then there exists $n_x \in \N$ such that $x \leq n_x$.
\end{theorem}

\begin{corollary}
	If $S:= \{1/n : n \in \N\}$, then $\inf S = 0$.
\end{corollary}

\begin{corollary}
	If $t >0$, there exists $n_t \in \N$ such that $0 < 1/n_t < t$.
\end{corollary}

\begin{corollary}
	If $y>0$, there exists $n_y \in \N$ such that $n_y -1 \leq y \leq n_y$.
\end{corollary}

\begin{theorem}
	There exists a positive real number $x$ such that $x^2 = 2$.
\end{theorem}

\begin{theorem}[\textbf{The Density Theorem}]
	If $x$ and $y$ are any real numbers with $x<y$, then there exists a rational number $r \in \Q$ such that $x < r < y$.
\end{theorem}

\begin{corollary}
	If $x$ and $y$ are real numbers with $x < y$, then there exists an irrational number $z$ such that $x < z < y$.
\end{corollary}
