**Theorems:**

1. A sequence is

**increasing**if $a_n$ < $a_{n+1}$ for every $n \geq 1$.

2. A sequence is

**decreasing**if $a_n$ > $a_{n+1}$ for every $n \geq 1$.

3. If a sequence is

**increasing**or

**decreasing**, then we call it

**monotonic**.

4. A sequence is

**bounded above**if there exists a number N such that $a_n \leq N$ for every $n \geq 1$.

5. A sequence is

**bounded below**if there exists a number M such that $a_n \geq M$ for every $n \geq 1$.

6. A sequence is

**bounded**if it is both

**bounded above**and

**bounded below**.

7. If the sequence is both

**monotonic**and

**bounded**, then it is always convergent.