Convergence of a sequence implies that it satisfies the Cauchy condition, namely . (Note that the Cauchy criterion does not imply convergence unless the metric space is complete.)
Now consider (1). We have whenever . Hence the Cauchy criterion with does not hold.
On the other hand, convergence of a sequence to is defined as saying that the sequence of real numbers converges to zero.
For (2), you have to decide what the sequence converges to, and I suggest you consider the sequence . Then consider to see that .