where the expression in the middle (the one with the dots ...) actually means (and not the sum to infinity which I originally took it to mean).
The reason I ask is that this ties in with the preamble to the question. For instance, look at the terms in the fourth set of brackets:
This corresponds to , , . So it is the sequence
There are terms in this bracket, each of which is less than the last term in the previous bracket, because for all , and so their sum is less than
Generalise this to , and you get
which I think is what you want for the right-hand side of the inequality. The left-hand side is similar, but not quite the same.
To look at the convergence of , I think you then use the fact that the sequence is squeezed in between the other two.
Does that give you some ideas?