Monotonic and bounded sequences
Monotonic and bounded sequences
Lessons
Notes:
Note *Theorem
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.

1.
Overview:

2.
Difference between monotonic and nonmonotonic sequences
Show that the following sequences is monotonic. Is it an increasing or decreasing sequence? 
3.
Difference between bounded, bounded above, and bounded below
Determine whether the sequences are bounded below, bounded above, both, or neither 
4.
Convegence of sequences
Are the following sequences convergent according to theorem 7?