Convergence in $L^p(\Omega)$ Norm

Question

Let $\Omega\subset\mathbb{R}^n$ be a bounded domain. Let $u,v,v_n\in L^p(\Omega)$ and suppose that $$\|u+v_n\|_p\rightarrow\|u+v\|_p$$

Is true that $$\|v_n\|_p\rightarrow\|v\|_p$$

Thanks

Answer 1

This is false. Consider the constant functions $u=-1,v_n=1+(-1)^n,v=0$.