Monotone Convergence Theorem

.Premise:

1. Function, ƒ, is bounded.

2. Function, ƒ, is monotone.

To prove:

ƒ converges.

Intuition:

ƒ is bounded so the range, S, of ƒ has an upper bound i.e. for y ∈ S ( y ≤ M).

Therefore, by Axiom of Completeness, S has a Least Upper Bound, s.

Proof of s = lim ƒ :

For e > 0, s-e is no longer Supremum (by Supremem Theorem). Therefore, for e > 0, there exists n ∈ N where s > ƒ(n) > s-e.

ƒ is monotone, so all m > n, s > ƒ(m) > e.

QED