# Note 3 - Sequences

## An Example

## Monotonicity

**Monotone Sequence**:

- () is increasing if
- () is decreasing if

Theorem: Any bounded monotone sequence converges.

Proof: bounded increasing. Let . We note that is bounded, so there exists .

Claim: .

Idea: We must find an such that contains .

- There is some such that
- This is true because if no such element exists, then must be . This contradicts the claim that .
- A consequence is that we know there exists at least one within .

- Then we will show .

Proof: , we claim that . Then, , we have . This implies that .

We note that sequences do not have to be monotone to converge.

converges to .

## Some Limit Theorems

If thm.

- If .

### Example

Let us take . We can divide top and bottom by . Then, the top and bottom are both sequences. We take of their limits.

### Example 2

Let us take:

**Proof that is bounded and increasing:**

- Bounded: .
- Increasing: