Question about convergence in $H^1_0$ Please how to prove that if $u_n\rightarrow u$ on $H^1_0$ we have that $||u_n||\rightarrow ||u||$ ? 
Please i need your help
Thank you 

In general, convergence in the norm implies convergence of the norms; that is, if $x_n \to x$ in the norm, then $\|x_n\| \to \|x\|$. This follows from the reverse triangle inequality:
$$\Big|\|x_n\| - \|x\|\Big| \le \|x_n - x\| \to 0$$
Alternatively, apply the triangle inequality to $x = (x - x_n) + x_n$ to get that $$\|x\| \le \liminf_{n \to \infty} \|x_n\|$$
and to $x_n = x_n - x + x$ to get that $\limsup_{n \to \infty} \|x_n\| \le \|x\|$. 
Note that this doesn't have to do with the particular norm in use here.
