Monotone Sequences and Convergence
A monotone sequence is either increasing so that ifthenor decreasing so that if thenWe make this rigorous in the following definition.
Definition A sequenceis increasing if and only iffor all positive integersA sequenceis decreasing iffor all positive integersA sequence is monotone if and only if it is either increasing or decreasing.
A monotone sequence is convergent if and only if it is bounded.
Proof: Supposeis a monotone sequence that is increasing and bounded above, thenis bounded above so letChoosethen sinceis the least upper bound ofis not an upper bound, so there is no such such thathenceconverges to
Supposeis a monotone sequence that is decreasing and bounded below, thenis bounded below so letChoosethen sinceis the greatest lower bound ofis not a lower bound, so there is nosuch thatForhenceconverges to