Let $x_1=(1,0,0,...),x_2=(1/2,1/2,0,0,...),x_3=(1/3,1/3,1/3,0,0,0,...),...$
Show that $x_n$ converges to zero in $\ell ^2$ and $\ell ^{\infty}$ spaces but it doesn’t converge in $\ell ^1$
I have written the sequence as
$(x_i)_n$ = \begin{cases} 0, & \text{if $ i\gt n$} \\ 1/n, & \text{if $i \le n$} \end{cases}
For $\ell ^{\infty}$ space :
$\|x_n-0\|_{\infty} = sup_{i \in \mathbb N}|x_{in}| = 1/n \lt \varepsilon$
For $\ell ^1$ space :
$\|x_n-0\|_{\ell ^1} = \sum_{i=1}^{\infty} |x_{ni}| = \sum_{i=1}^{\infty} 1/n = 1/n \lt \varepsilon $ since $n$ is independent from index $i$. Thus it converges to zero.
I confused about this. How must I write for $\ell ^1$ space? I cannot see my mistake. Thanks...
$|x_n|_{\ell^1}=\frac{1}{n}+\cdots+\frac{1}{n}=1\not\to 0$, while
$|x_n|_{\ell^2}=\left(\frac{1}{n^2}+\cdots+\frac{1}{n^2}\right)^{1/2}=\frac{1}{n^{1/2}}\to 0$.