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\begin{center}\textbf{Problem Set 2 \hspace{.7in} Real Analysis 1 \hspace{.7in} Due 9/9/18} \end{center}

\noindent Four of these problems will be graded, with each problem worth 5 points.  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 Let $A, B \in \mathbb R$. Show that $\forall \epsilon > 0, |A - B| < \epsilon  \Rightarrow A = B$ .

\item Suppose that a set $S$ has an infimum.  Show that $\inf S$ is unique.

\item Show that if a sequence is convergent, then it is bounded.

\item Suppose that $\{a_n\}_{n=1}^\infty$ is a convergent sequence.  Is  $\{a_{2n}\}_{n=1}^\infty$ convergent as well?


\item The sequence $\{a_n\}$ converges to 0 if and only if the sequence $\{|a_n|\}$ converges to 0.

\item The sequence $\{a_n\}$ converges to $A$ if and only if the sequence $\{|a_n|\}$ converges to $|A|$.

\item Let $\{a_n\}$ be defined by $\frac{5n}{2n+1}$ for $n \in \mathbb N$. Show that this sequence converges

\item If $\{a_n\}$ and $\{b_n\}$ differ from each other in only a finite number of terms, then both sequences converge to the same value or they both diverge. [Kosmala 2.1.8(b)]

\end{enumerate}

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