Some definitions:

  • A sequence is bounded if $|a_n|$ never grows beyond a fixed size $M$. In other words, there is a bound $M$ such that every term in the sequence has size less than $M$.

  • A sequence {$a_n$} is strictly increasing if each term is bigger than the previous term. That is, $a_{n+1} > a_n$. It is non-decreasing if $a_{n+1} \ge a_n$.

  • Strictly decreasing means $a_{n+1}< a_n$ for all $n$, and non-increasing means $a_{n+1} \le a_n$.

  • If a sequence is either non-increasing or non-decreasing, it is called monotonic.

Caution: The terms increasing and decreasing are dangerously ambiguous, since some authors use them to mean "strictly increasing" and "strictly decreasing", while others use them to mean "non-decreasing" and "non-increasing".

Two important Theorems:

  • Monotonic Convergence Theorem: If a sequence is monotonic and bounded, if converges.

  • Theorem: If a sequence is not bounded, it diverges.

Caution: If a sequence is bounded but not monotonic, it might converge and it might diverge. For example,