For the first part:

pick anx. The sequence is non-zero at most 1 time. So there exists somemsuch that for . So the tail of the sequence converges to 0, since it is identically 0, and thus so does the whole sequence.

For the second part, look at the limit in 3 cases:

For since for alln.

For since again for a givenxthere is somemat which for alln>m.

For since for alln.