Number of limit points of a sequence

Question

$x_n$ be a real sequence converging to $x$. What is the maximum number of limit points of the sequence $y_n=\lfloor {x_n^2} \rfloor+\lfloor {x_n} \rfloor$?

Answer 1

I attempted this question on the basis of where the values of $x_n$ lie.

Case 1) If $x_n$ is a integer sequence then it is eventually constant and so is the sequence $y_n$.

Case 2) If $x_n$ is not an integer sequence then we take two cases- $x\in \mathbb{Z}$ and $x\notin \mathbb{Z}$. In the later case there will be nbds $B(x,\epsilon_1)$ and $B(x^2,\epsilon_2)$ such that $B(x,\epsilon_1) \subset [k,k+1]$ and $B(x^2,\epsilon_2)\subset[m,m+1]$, i.e., after finite number of terms the sequences $x_n$ and $x_n^2$ will be properly contained in the closed interval with integer boundary points. Hence, $y_n\to k+m$ as $n\to \infty$. In the former case there can be further two cases, $x_n$ convrging to $x$ from one side or it converges as an alternating sequence. In these cases proceeding similarly we get number of possible limit points of the sequence $y_n$ equal to 1 and 2 respectively.