I have 2 questions about precompactness in sequence spacesDoh)

1. Is A = { $\displaystyle (x_{n} : \sum_{n=1}^{\infty} \sqrt{n} |x_{n} | \leq 1 $ } is precompact in $\displaystyle c_{0}$ = set of all sequences converging to 0 equipped with the sup norm ?

2. Is A= { $\displaystyle (x_{n} : n x_{n} \rightarrow 0 $ as $\displaystyle n \rightarrow \infty$ } in $\displaystyle l_{1} $ ?