The image shows a mathematical exercise asking to write statements using quantifiers about a sequence $(u_n)_{n \in \mathbb{N}}$. The six propositions refer to different properties of the sequence such as increasing, decreasing, boundedness, and constancy.

Here is how to express these statements using quantifiers:

1. La suite $(u_n)_{n \in \mathbb{N}}$ est croissante (The sequence is increasing): $\forall n \in \mathbb{N},\ u_{n+1} \geq u_n$

2. La suite $(u_n)_{n \in \mathbb{N}}$ est décroissante (The sequence is decreasing): $\forall n \in \mathbb{N},\ u_{n+1} \leq u_n$

3. La suite $(u_n)_{n \in \mathbb{N}}$ est minorée (The sequence is bounded below): $\exists m \in \mathbb{R},\ \forall n \in \mathbb{N},\ u_n \geq m$

4. La suite $(u_n)_{n \in \mathbb{N}}$ est majorée (The sequence is bounded above): $\exists M \in \mathbb{R},\ \forall n \in \mathbb{N},\ u_n \leq M$

5. La suite $(u_n)_{n \in \mathbb{N}}$ est bornée (The sequence is bounded): $\exists m, M \in \mathbb{R},\ \forall n \in \mathbb{N},\ m \leq u_n \leq M$

6. La suite $(u_n)_{n \in \mathbb{N}}$ est constante (The sequence is constant): $\exists c \in \mathbb{R},\ \forall n \in \mathbb{N},\ u_n = c$

Would you like more details on any of these expressions?

Here are 5 related questions you might consider:

1. What are the implications of a sequence being both bounded above and below?
2. How does a constant sequence relate to bounded and monotonic sequences?
3. Can an unbounded sequence still be increasing or decreasing? Why or why not?
4. How would you describe a strictly increasing or strictly decreasing sequence using quantifiers?
5. What is the relationship between a bounded sequence and its limit points?

Tip: Quantifiers are key in formalizing mathematical properties. "∀" means "for all" and "∃" means "there exists."