Let $B_R$ be the ball (centered in $0$) of radius $R>0$ and let $\varphi\in C_C^{\infty}(B_R)$, where $C_C^{\infty}(B_R)$ stands for the set of $C^{\infty}$ with compact support functions. Moreover, let $(v_k)_k$ be a bounded sequence. My question is: Can I conclude that $$\int_{B_R} v_k^2 |\nabla\varphi| dx \le c,$$ where $c>0$ denotes a constant?
About me the answer is yes, because $|\nabla\varphi|$ is continuous over a compact set so it is bounded and $v_k^2$ is bounded since it is $v_k$. Could someone please tell me if I am right or I miss something?
Thank you in advance!