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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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    Are you sure this is what you want to ask? Convergence *of* norm and convergence *in* norm are different things...2012-11-10
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    Yes it is @MihaHabič2012-11-10

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