Bounded sequences.

Definition 3.2.1   Let $(u_n)$ be a sequence.

1. The sequence $(u_n)$ is bounded above if there exists a real number $M$ such that $\forall n \in \mathbb{N}, \; u_n \leq M$ .
2. The sequence $(u_n)$ is bounded below if there exists a real number $m$ such that $\forall n \in \mathbb{N}, \; u_n \geq m$ .
3. The sequence $(u_n)$ is bounded if it is bounded above and bounded below.

Example 3.2.2   Let $u_n=n^2$ for any $n \in \mathbb{N}$ .

• The sequence $(u_n)$ is bounded below:

  $$\forall n \in \mathbb{N}, \; u_n \geq 0.$$

  
  

• The sequence $(u_n)$ is not bounded above, i.e. for any real positive number $A$ , there exists an $n \in \mathbb{N}$ such that $u_n > A$ . We have:

  $$u_n >A \Longleftrightarrow n^2 >A \underbrace{\Longleftrightarrow}_{\text{both sides are positive}} n > \sqrt{A}.$$

  
  
  Actually, for any $n> [\sqrt{A}]$ , we have $u_n > A$ .

Example 3.2.3   The sequence whose general term is $\sin \left( \frac {n!\vert 2n}{\sqrt{n^3+n!}} \right)$ is bounded, as everyu sine is greater than or equal to -1 and less than or equal to 1.