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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

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    Yes it is @MihaHabič2012-11-10

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