Thank you all,
this excercise is solved.
and is 0 as soon as by definition. However, is defined in such a way that it is smaller than the space between two values of the sequence (which is increasing), so that for any , and finally for any . We conclude that the only non-zero term in the previous sum is . Remember and you are done.
In case 1), there seems to be an implicit assumption here that is an increasing sequence of real numbers. If so, then the condition implies that whenever . But you are given that whenever , and therefore whenever . Hence (because the only nonzero term in the sum is the one where m=n). So on N.
The data for case 2) looks wrong. In fact, the two conditions and if will contradict each other if a<2. And there is nothing to indicate what a is supposed to be.
Edit. Beaten to it by Laurent!