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\begin{center}\textbf{Problem Set 4 \hspace{.7in} Real Analysis 1 \hspace{.7in} Due 9/20/24} \end{center}

\noindent Five of these problems will be graded, with each problem worth 4 points. \LaTeX is expected.  Clear and complete justification is required for full credit. You are welcome to discuss these problems with anyone and everyone, but must write up your own final submission without reference to any sources other than the textbook and instructor.

\begin{enumerate}
\item If a sequence $\{ a_n \}$ is bounded and all of its convergent subsequences have the same limit, then $\{ a_n \}$ converges.
 
\item Define a sequence recursively by letting $a_0 = 0$ and letting $a_{n+1} = \sqrt{2+a_n} \: \forall n \in \mathbb{N}$.  Determine whether $a_n$ converges.

\item Suppose that $\lim_{x \to \infty} f(x) = A$, where $f$ is a function with domain $D$.  Prove (directly from the definition) that if $k$ is a constant, $\lim_{x \to \infty} [k f(x) ] = k A $.

\item Suppose that $\lim_{x \to \infty} f(x) = A$, where $f$ is a function with domain $D$.  Prove (directly from the definition) that $ \lim_{x \to \infty} \left | f(x) \right | =  \left | \lim_{x \to \infty}  f(x) \right |$.

\item Suppose that $\lim_{x \to \infty} f(x) = A$ and $\lim_{x \to \infty} g(x) = B$, where $f$ and $g$ are functions with domain $D$.  Prove (directly from the definition) that $\lim_{x \to \infty} [f(x) \cdot g(x)] = A \cdot B$.

\item If a limit exists for $f$ as $x$ approaches $a$, then that limit is unique.

\item Suppose that $\lim_{x \to \infty} f(x) = A$ and $\lim_{x \to \infty} g(x) = B$, where $f$ and $g$ are functions with domain $D$. If $\forall x \in D, f(x) < g(x)$ then $A < B$.

\item Suppose that $\lim_{x \to \infty} f(x) = A$ and $\lim_{x \to \infty} g(x) = B$, where $f$ and $g$ are functions with domain $D$. If $\forall x \in D, f(x) \leq g(x)$ then $A \leq B$.

\item If $f:D \to \mathbb{R}$ and $\lim f(x)_{x \to a}$ exists, then $f$ is bounded.

\item If $f:D \to \mathbb{R}$ and $\lim f(x)_{x \to a}$ exists, then $f$ is bounded on some set $D_1$, with $D_1 \subseteq D$.

\end{enumerate}

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