Exercise 10

Let \( P \) be the following proposition: \( \forall n \in \mathbb{N}, \exists m \in \mathbb{N} \) such that \( m < n \).

1. Show that \( P \) is false.

Let \( f \) be the function defined on \( \mathbb{R} \) by:

\( f(x) = x^2 - 5x + 4 \)

2. Show that \( f \) is not even.

3. Show that \( f \) is not odd.

4. Show that \( f \) is not decreasing on \( \mathbb{R} \).

Exercise 11

1. Show that: \( \forall n \in \mathbb{N}^* \), \[ \sum_{k=1}^n \frac{1}{k^2} \leq 2 - \frac{1}{n} < 2 \]
2. Show that: \[ \forall n \in \mathbb{N}, \exists a_n, b_n \in \mathbb{N} : \begin{cases} (1 + \sqrt{2})^n = a_n + b_n \sqrt{2}, \\ a_n^2 - 2b_n^2 = (-1)^n. \end{cases} \]
3. Show that: \( \forall n \in \mathbb{N}^* \), \[ 1 + \frac{1}{\sqrt{2}} + \cdots + \frac{1}{\sqrt{n}} \geq \sqrt{n}. \]