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\noindent {\bf Question}

\noindent What can be said about a sequence $\{ a_n\}$ if it
converges and if every $a_n$ is an integer?  Also, give a
qualitative description of all of the convergent subsequences of
the sequence
\[ 1, 1, 2, 1, 2, 3, 1, 2, 3, 4, 1, 2, 3, 4, 5, \ldots.\]

\medskip

\noindent {\bf Answer}

\noindent A convergent sequence of integers must be eventually
constant; that is, there exists $M$ so that $a_n =a_p$ for all
$n$, $p
>M$.  This follows from the Cauchy criterion with $\varepsilon
=\frac{1}{2}$ and the fact that the difference of two non-equal
integers is at least $1$.

\medskip
\noindent For this given sequence, the convergent subsequences are
all of the following form: pick a positive integer $p$, and note
that $p$ appears infinitely many times in the given sequence.
Then, a convergent subsequence is of the form $a_0, a_1, \ldots,
a_M, a_{M+1} =p, a_{M+2} =p, \ldots$ for some $M$, where
$a_0,\ldots, a_M$ are arbitrary positive integers.


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