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 .

  1. 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 .
  2. 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.

  1. 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: