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}$ .

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.

Noah Dana-Picard 2007-12-28