Question 180
Let $(x_n)_{n \ge 1}$ be a sequence of real numbers from $[1, \infty)$. It is known that the sequences $(y_n^{(k)})_{n \ge 1}$, defined by $y_n^{(k)} = \lfloor x_n^k \rfloor$, $n \ge 1$, are convergent for every $k \in \mathbb{N}^*$. Prove that the sequence $(x_n)_{n \ge 1}$ is convergent.