Monotonous sequences.

Definition 3.5.1   Let $(u_n)$ be a sequence of real numbers.

1. The sequence $(u_n)$ is constant if $\forall n, \; u_{n+1}=u_n$ .
2. The sequence $(u_n)$ increases if $\forall n, \; u_{n+1} \geq u_n$ .
3. The sequence $(u_n)$ decreases if $\forall n, \; u_{n+1} \leq u_n$ .
4. The sequence $(u_n)$ is monotonous if it is either decreasing or increasing.

Example 3.5.2       

• An arithmetic sequence whose difference is positive is an increasing sequence.
• An arithmetic sequence whose difference is negative is a decreasing sequence.
• A geometric sequence whose quotient is greater than 1 and whose first term is positive is an increasing sequence.
• A geometric sequence whose quotient is negative is not monotonous.
• The sequence whose general term is equal to $n!$ is an increasing sequence.
• The number of atoms of Uranium 235 present in an atomic pile, checked every minute, define a decreasing sequence.