Examples 2.8, 2.9, and 2.10 are all series whose partial sums are monotonically increasing sequences. 2.8 and 2.10 are bounded, hence converge. 2.9, the harmonic series is not bounded, hence does not converge. Subsequence A subsequence is obtained from a sequence by extracting from this sequence only certain elements and dropping the remaining ones. See the notation in your text, page 37.
Every sequence has a monotone subsequence.
Our second chosen element is also not a peak index, so we can repeat this process again to choose the third element for our sequence. We recursively repeat this process to define the subsequence. Just construct the subsequence consisting of the elements with peak indices. The previous theorem tells us we can construct a subsequence which is monotone. It will also be bounded.