The monotone convergence theorem (MCT) states that a nondecreasing sequence of reals that is bounded from above has a least upper bound.

The MCT can be used when analyzing sums of the terms of infinite series. Consider a the $S_n = \sum_{i=a}^{n}{ s_i }$. If the terms of $s_i$ are all nonnegative, then the terms of the sequence $S_n$ are a nondecreasing sequence. If the sequence $S_n$ does not diverge, then it has a least upper bound by the MCT. This leads to the conclusion that the $\lim_{n \to \infty}{ \sum_{i=a}^{n} s_i }$ exists.