Proposition 3.1.9: Monotone Sequences
If is a monotone increasing sequence that is bounded above, then the sequence must converge (see picture).
If is a monotone
decreasing sequence that is bounded below, then the sequence must converge
Proof:Let's look at the first statement, i.e. the sequence in monotone increasing. Take an > 0 and let c = sup(xk). Then c is finite, and given > 0, there exists at least one integer N such that xN > c - . Since the sequence is monotone increasing, we then have that
xk > c -for all k > N, or
| c - xk | <for all k > N. But that means, by definition, that the sequence converges to c.
The proof for the infimum is very similar, and is left as an exercise.