$\DeclareMathOperator\Lip{Lip}$This question arose when I read Godefroy and Lerner - Some natural subspaces and quotient spaces of $L^1$.

Let $\Lip_0(\mathbb R^n)$ be the space of Lipschitz functions $f:\mathbb R^d\to\mathbb R$ vanishing at the origin, $f(0)=0$. It is known from the above paper that, endowed with the norm $\|f\|_{\Lip}\mathrel{:=}\|\nabla f\|_{\infty}$, $\big(\Lip_0(\mathbb R^n), \|\cdot\|_{\Lip}\big)$ is a Banach space. My question is, if $f^n$ converges to $f$ under the above norm, could we deduce $$\lim_{n\to\infty} \int_{\mathbb R^d}\big(f^n(x)-f(x)\big)u(x)dx = 0 ,$$ where $u:\mathbb R^d \to\mathbb R_+$ is a measurable function s.t. $$ \int_{\mathbb R^d}(1+\lvert x\rvert)u(x)dx <\infty.$$

This seems a trivial question, but I can not prove it rigorously for general dimensions. Is there any classical reference?