 Solution

The floor function has even value on indices congruent to zero or one modulo four, and odd value on indices congruent to two or three. Accordingly the displayed sequence has terms $2+1/n$ in the first two residue classes and $-1/n$ in the other two. In particular,Every term is within $1/n$ of the set $\{0,2\}$. If a subsequence converged to $c\notin \{0,2\}$, its eventual distance from that set would be bounded below by a positive constant, contradicting this bound. The complete set of subsequential limits is $\{0,2\}$.