Proof of alternating series test

Hi,

I'm having trouble understanding the first line of Knapp's proof of the alternating series test. The Theorem states that

If for each in a nonempty set S, is a monotone decreasing sequence of nonnegative real numbers such that uniformly in , then converges uniformly.

His opening line in the proof is

The hypotheses are such that whenever ...

It's the above line I'm totally confused by. Can someone help explain to me how the hypothesis implies the above?

Thanks :)