Monotonic Sequence Convergence Theorem, Part 2: A bounded, monotonic decreasing sequence is convergent.
Proof: Let be a monotonic decreasing sequence, . is a bounded set of real numbers, thus, via the completeness axiom of real numbers, it must have an infimum. We call the infimum . Thus for all . We want to show that
Let . . For and . Thus, for all , which implies that converges to .
Note, part 1 of the proof can be found here.