1. Theorem Suppose {a n } is non-decreasing and bounded above by a number A. Then {a n } converges to some finite limit a, with a  A. Suppose {b n } is non-increasing and bounded below by a number B. Then {b n } converges to some finite limit b, with b  B.

2. Boundedness A sequence is bounded provided there is a positive number K such that |a n |  K for all values of n.

3. Monotonicity A sequence {a n } is monotone if either a) a n  a n+1 or b) a n  a n+1 for all n. The sequence is non-decreasing if situation a holds, and non-increasing if situation b holds.

4. Proving Monotonicity: Part 1 Algebraic Difference: 1) If a n+1 – a n  0 for all n, then the sequence is non-increasing. 2)If a n+1 – a n  0 for all n, then the sequence is non-decreasing. Example Show that the sequence is monotone.

5. Proving Monotonicity: Part 2 Algebraic Ratio: If a n > 0 for all n, then 1) If a n+1 /a n  1 for all n, the sequence is non-increasing. 2)If a n+1 /a n  1 for all n, the sequence is non-decreasing. Example Show that the sequence is monotone.

6. Proving Monotonicity: Part 3 Derivative: If a n = f (n), then 1) If f (n)  0 for all n, the sequence is non-increasing. 2)If f (n)  0 for all n, the sequence is non-decreasing. Example Show that the sequence is monotone.