## Problem 4.

Let $\left(x_{n}\right)_{n \geq 1}$ be a sequence of real numbers with the property that

$$
\sqrt{n \cdot x_{n}+n+1}-\sqrt{n \cdot x_{n}+1} \leq \frac{\sqrt{n}}{2} \leq \sqrt{n \cdot x_{n}+n}-\sqrt{n \cdot x_{n}},(\forall) n \geq 1
$$

Calculate the limit of the sequence $\left(x_{n}\right)_{n \geq 1}$.
## Solution and Scoring:

![](https://cdn.mathpix.com/cropped/2024_06_07_0026720ce8590832c3dbg-3.jpg?height=395&width=1587&top_left_y=1242&top_left_x=297)
$\sqrt{1+x_{n}+\frac{1}{n}} \leq \frac{1}{2}+\sqrt{x_{n}+\frac{1}{n}} \Rightarrow \frac{9}{16} \leq x_{n}+\frac{1}{n}, \forall n \geq 1$ ..... $2 p$
$\frac{1}{2}+\sqrt{x_{n}} \leq \sqrt{1+x_{n}} \Rightarrow x_{n} \leq \frac{9}{16}, \forall n \geq 1$ ..... $2 p$
$0 \leq \frac{9}{16}-x_{n} \leq \frac{1}{n}$, thus the sequence $x_{n}$ converges to $\frac{9}{16}$ ..... $2 p$