how to prove that:
every sequence which convergences has to be bounded
A sequence $\displaystyle \{s_n\}$ converges means that there exists an $\displaystyle N$ and an $\displaystyle s$ such that for all $\displaystyle n>N$
$\displaystyle |s_n-s|<1$.
Hence for all $\displaystyle n>N$:
$\displaystyle
|s_n|-|s|\le |s_n-s| < 1
$
so $\displaystyle |s_n|<1+|s|$
Therefore
$\displaystyle |s_n|\le \max \left[\left(\max_{0<k<N-1}|s_k|\right),1+|s|\right]$
CB