It's a result of Dirichlet's test: if is a bounded series (i.e., its partials sums sequence is bounded) and if is a descending monotone sequence that converges to zero, then converges:

Since is monotone and bounded it converges to a finite limit, say L. Assume it is monotone ascending (if it is descending it is very simmlilar) , so we get monotonically descending and is bounded because it is convergent, and thus by Dirichlet's test the series converges, but then:

is the difference of two convergent series and thus it converges, and by arithmetic of limts, and we're done.

Tonio