monotonic upper bound and lower boundgiven any

Question

Monotonic, Upper bound and lower bound

Given any sequence {a_n} we have the following terminology:

1.   We call or denote the sequence increasing if a_n < a_n+1 for every n.

2.   We call or denote the sequence decreasing if a_n > a_n+1 for every n.

3.   If {a_n} is an increasing sequence or {a_n} is a decreasing sequence we denote it monotonic.

4.   If there exists a number m like m ≤ a_n for each n we say the sequence is bounded below. Occasionally the number m is called a lower bound for the sequence.

5.   If there exists a number M like a_n  ≤ M for every n we say the sequence is bounded above. Occasionally the number M is called an upper bound for the sequence.

6.   If the sequence is both of the bounded below and bounded above we denote the sequence bounded.