# Monotonic and bounded sequences

##### Intros

##### Examples

###### Lessons

**Difference between monotonic and non-monotonic sequences**

Show that the following sequences is monotonic. Is it an increasing or decreasing sequence?**Difference between bounded, bounded above, and bounded below**

Determine whether the sequences are bounded below, bounded above, both, or neither**Convegence of sequences**

Are the following sequences convergent according to theorem 7?

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###### 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

1. A sequence is

2. A sequence is

3. If a sequence is

4. A sequence is

5. A sequence is

6. A sequence is

7. If the sequence is both

**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

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