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\noindent {\bf Question}

\noindent Prove that if a sequence $\{ a_n\}$ is increasing and
bounded above, then it is convergent.

\medskip

\noindent {\bf Answer}

\noindent Since $\{ a_n\}$ is bounded above, it has a supremum
$a$.  By the definition of supremum, for every $\varepsilon >0$,
there exists $M$ so that $| a_M -a |  <\varepsilon$.  Since $\{
a_n\}$ is increasing and since $a$ is an upper bound for $\{
a_n\}$, we have that $a_M <a_n\le a$ for every $n >M$.  In
particular, we have that $| a_n -a| < |a_M -a| < \varepsilon$ for
every $n >M$, and this is just the definition that $\{ a_n\}$
converges to $a$.


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