Q:

Is $l_p$ a closed subspace of $c_0$?

I would think it is, but I am not sure. 
In an attempt trying to prove it, I take a series $(a_n)_m:\mathbb{N}\rightarrow l_p$, which is convergent in $c_0$. My goal is to show that $\lim_{m\rightarrow\infty}(a_n)_m=:a_n \in l_p \iff (\sum^\infty_{n=0}|a_n|^p)^{\frac{1}{p}}<\infty$.
The convergence on $c_0$ means that for $\varepsilon \in \mathbb{R}$ $\exists N \,\, \forall k\in \mathbb{N}: $
$$\operatorname{sup}_{n\in \mathbb{N}}|a_n-(a_n)_k|<\varepsilon$$.
I'm not sure how to go on. Can you help me, please? 
Thanks in advance! 

A:

No. Let $a=(k^{-1/p})_k$ and
$$
a^n=(1,2^{-1/p},\dots,n^{-1/p},0,0,0,\dots).
$$
Then $a^n$ converges to $a$ in $c_0$, but $a\notin\ell^p$.

