Give an example of a sequence in $l^{1}$ that:
1) converges a zero in $l^{\infty}$ but is not bounded in $l^{2}$;
2) converges a zero in $l^{2}$ but is not bounded in $l^{1}$.
Can anyone help me?
I thought to use these sequences
$$1) x(n)=\frac{e_n}{n^{1/3}}$$
$$2) x(n)=\frac{1}{n}$$
Are they correct?
For 1) define $a_{n,j}=\frac 1 n$ for $j\leq n^{3}$ and $0$ otherwise. For 2) define $a_{n,j}=\frac 1 j$ for $n\leq j\leq n^{2}$ and $0$ otherwise. The sequences $a_n=(a_{n,j})$, $n=1,2,\cdots$ have the required properties. For 2) you need the fact that $\sum_n^{n^{2}}\frac 1 j \to \infty$. For this compare with the integral $\int_n^{n^{2}} \frac 1 x\, dx$ which is $\log\, n$.