## Problem 2:

a) Study whether there exists a non-monotonic sequence $\left(x_{n}\right)_{n \geqslant 1}$ such that $\lim _{n \rightarrow \infty} x_{n}=\infty$.

b) Consider the sequence $\left(x_{n}\right)_{n \geq 1}$ for which $x_{n}^{2}-2 n x_{n}+n^{2}-1 \leq 0, \forall n \geq 1$. Calculate $\lim _{n \rightarrow \infty} \frac{x_{n}}{n}$.

RMCS 36
Problem 2:

From $\left(x_{n}-n\right)^{2} \leq 1$ we deduce $n-1 \leq x_{n} \leq n+1$. 3pts

Apply the squeeze theorem and obtain $\lim _{n \rightarrow \infty} \frac{x_{n}}{n}=1$. 4pts