I am having troubles proving the following theorem. Suppose that a sequence of positive numbers satisfies for all n:

then this sequence converges.

I was given the hint

I tried plugging this hint, and tried to showed that If the sequence is monotone after some term, then the series converges since it is a bounded sequence. However, if this series continues to oscilate indefinitely, I haven't been able to show that it converges.

help please!