let $\{p_n\}$ be a sequence of integers. prove that if $\{p_n\}$ is cauchy, then $\{p_n|n \in N\}$ is finite. my proof: don't know what else do it need or is it right please help...
pf: $Given, \{p_n\}$ is sequence of integers which is cauchy.
By definition of cauchy sequences,
For every $\epsilon>0,$ there exists N $\in \mathbb{N}$ such that for $m,n \in \mathbb{N}$ $ and$ $ m,n \geq N$
$|p_m-p_n|<\epsilon$
For $ \epsilon $= $\frac{1}{2}$$ ,$ we have$ $ $|p_m-p_n|$ $<\frac{1}{2} .$
Since $p_m$ and $p_n $are integers, $|p_m-p_n|=0$ $ or$ $ p_m=p_n$ for all $ m,n \geq N$ for some $N \in$ $\mathbb{N} .$
All the terms after this are the same as and equal $p_m=p_n=p_N.$
Therfore, the set $\{p_n | n \in \mathbb{N}\} $ is finite and it contains atmost N elements namely $\{p_1,p_2,...,p_N\}$
(atmost because there might exist $ i,j<N $such that$ p_i=p_j$ $ i.e.$ elemets are repeated.