Convergence in $\ell^p$ norm provided it weakly converges. I need some help with the following problem : 

$1<p < \infty$ , let $x_n$ be a sequence in $\ell^p$ and also $x\in \ell^p$ . I am interested in showing $$\lim_{n\to \infty} \|x_n-x\|_p\to0$$ if and only if $x_n$ converges to $x$ in weak topology and $$\lim_{n\to \infty} \|x_n\|=\|x\|_p$$

I need someone to show me the footsteps that I should follow and think . 
Thank you . 
 A: It's a general fact that strong convergence implies weak convergence and convergence of the norms. 
To see the converse, fix $\varepsilon>0$. There is an integer $N$ such that $\sum_{k\geqslant N}|x^k|^p<\varepsilon$. Weak convergence gives convergence of the coordinates (i.e. $x_n^k\to x^k$ for each $k$, using the linear functional $x\mapsto x^k$), so we also have for $n$ large enough that $\sum_{k\geqslant N}|x_n^k|^p<\varepsilon$. Indeed, we have $\sum_{k=1}^{N-1}|x_n^k|^p\to \sum_{k=1}^{N-1}|x^k|^p$ by weak convergence, and $\sum_{k\geqslant 1}|x_n^k|^p\to \sum_{k\geqslant 1}|x^k|^p$, which gives $\sum_{k\geqslant N}|x_n^k|^p\to \sum_{k\geqslant N}|x^k|^p$. 
This gives, by the inequality $|a+b|^p\leqslant 2^{p-1}(|a|^p+|b|^p)$, 
$$\lVert x_n-x\rVert_p^p\leqslant \sum_{j=1}^{N-1}|x_n^j-x^j|^p+2^{p-1}\cdot 2\varepsilon.$$
Taking the $\limsup_{n\to +\infty}$, and using the fact that $\varepsilon$ is arbitrary, we get the wanted result. 



*

*Note that the result also holds when $p=1$, and actually, we just need weak converge. It's done in this thread.

*We can use Fatou's lemma applied to counting measure and the sequence $y_n^k:=2^{p—1}(|x_n^k|+|x^k|)-|x_n^k-x^k|$.

