Consider the following properties of a sequence $\left\{a_{n}\right\}_{n}$ of real numbers.
$\text{(I)}\displaystyle{} \lim _{n \rightarrow \infty} a_{n}=0$.
$\text{(II)}$ There exists a sequence $\left\{i_{n}\right\}_{n}$ of positive integers such that $\sum_{n=1}^{\infty} a_{i_{n}}$ converges.
Which of the following statements is correct?
- $\text{(I)}$ implies $\text{(II)},$ and $\text{(II)}$ implies $\text{(I)}$.
- $\text{(I)}$ implies $\text{(II)},$ but $\text{(II)}$ does not imply $\text{(I)}$.
- $\text{(I)}$ does not imply $\text{(II)},$ but $\text{(II)}$ implies $\text{(I)}$.
- $\text{(I)}$ does not imply $\text{(II)},$ and $\text{(II)}$ does not imply $\text{(I)}$.
