- () is increasing if
- () is decreasing if
Theorem: Any bounded monotone sequence converges.
Proof: bounded increasing. Let . We note that is bounded, so there exists .
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 .
Let us take . We can divide top and bottom by . Then, the top and bottom are both sequences. We take of their limits.
Let us take:
Proof that is bounded and increasing:
- Bounded: .