Proposition: 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 (see picture).

    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.

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    (bgw)