Monotonic and bounded sequences

Topic Notes

In this section, we will be talking about monotonic and bounded sequences. We will learn that monotonic sequences are sequences which constantly increase or constantly decrease. We also learn that a sequence is bounded above if the sequence has a maximum value, and is bounded below if the sequence has a minimum value. Of course, sequences can be both bounded above and below. Lastly, we will take a look at applying theorem 7, which will help us determine if the sequence is convergent. One important to note from the theorem is that even if theorem 7 does not apply to the sequence, there is a possibility that the sequence is convergent. It's just that the theorem will not be able to show it.

Note

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.

Basic Concepts

Introduction to sequences