This image contains a set of mathematical questions, likely from an exercise on sequences and limits. I will help you step by step to solve each one. Let's break down the tasks:

1) Show that for all $n \in \mathbb{N}$, $U_n \geq 0$

This part is asking to prove that the terms of the sequence $(U_n)$ are non-negative for all natural numbers $n$.

2) Consider the sequence $(V_n)$ defined for $n \in \mathbb{N}$ as $V_n = \sqrt{n} + U_n$

a) Express $V_{n+1} - V_n$ in terms of $n$ and $U_n$ b) Deduce that $(V_n)$ is decreasing and that for all $n \in \mathbb{N}, U_n \leq \frac{1}{2\sqrt{n}}$

3) Consider the sequence $(W_n)$ defined for $n \in \mathbb{N}$ as $W_n = \sqrt{n} U_n$

a) Prove that $(W_n)$ is increasing. b) Deduce that for all $n \in \mathbb{N}, U_n \geq \frac{1}{4\sqrt{n}}$

4) Show that the sequence $(U_n)$ is convergent and compute its limit.

5) a) Calculate $U_n$ as a function of $n$.

b) Deduce that $\lim_{n \to \infty} \frac{(2n)!}{n^{2n}}$ exists.

I can guide you through the solutions step-by-step. Would you like to start with the first question?