The Monotone Sequence Theorem says that all monotonic sequences which are bounded will converge and the limit is their least upper bound or greatest lower bound (depending on the direction of the sequence).
In layman's terms this means that if you have a series of numbers which always gets bigger of small (thats what monotonic means) e.g.
and you know that they never exceed a certain number, e.g. in this case they never get less that 0, or -1, or -2, ...). Then if you keep on down the sequence you will eventually (in a mathematical sense) reach a limit and this limit is the smallest number the sequence is bigger than, (or biggest number the sequence is smaller than if the sequence is decreasing).