Created the Real Analysis Theorems and Definitions packet

chapter-2/applications-of-the-supremum-property.tex

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+\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}