Question 148
Let $\left(x_{n}\right)_{n \geq 1}$ be a sequence of real numbers from the interval $[1, \infty)$. Suppose that the sequence $\left(y_{n}^{(k)}\right)_{n \geq 1}$, defined by $y_{n}^{(k)}=\left[x_{n}^{k}\right]$, $n \geq 1$, is convergent for every $k \in \mathbb{N}^{*}$. Prove that the sequence $\left(x_{n}\right)_{n \geq 1}$ is convergent. (By $[a]$ we denote the integer part of the real number $a$.)