Sequences and Series, A Bounded Increasing Sequence Converges
A Bounded Increasing Sequence Converges
If a monotonic function or sequence f() is bounded above, its range has a least upper bound, call it u.
For every ε there is some x with u-f(x) < ε,
else u would not be the least upper bound.
Thereafter, f is within ε of its limit u,
hence f converges to u.
Similarly, a monotonically decreasing function bounded below converges to its greatest lower bound.
If the terms of a series are all nonnegative,
the sequence of partial sums is monotonically increasing.
If this sequence is also bounded then the series converges.