Monotone Convergence Theorem
1. Function, ƒ, is bounded.
2. Function, ƒ, is monotone.
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.