= 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,
$$
\boxed{a_{4k}=2+\frac1{4k}\longrightarrow2,\qquad
a_{4k+2}=-\frac1{4k+2}\longrightarrow0.}
$$
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. \b[The complete set of subsequential limits is $\{0,2\}$.]
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