Any help please?
if my intuition is correct without any solid proof the sequences in concern are:
$\displaystyle \frac{2+(-1)^n}{n}$ and $\displaystyle \frac{2+(-1)^n}{n}+1$
or the sequences $\displaystyle \frac{3}{2k}+1$ and $\displaystyle \frac{1}{2k+1}$ which are the two subsequences of the major sequence
$\displaystyle S = \left\{ {\frac{{2 + \left( { - 1} \right)^n }}
{n}:n \in \mathbb{Z}^ + } \right\},~\inf (S) = 0,~\sup (S) = \frac{3}{2}$.
Clearly there are subsequences converging to 0 but none converging to $\displaystyle \frac{3}{2}$.