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Here is an alternative answer (also based on the control of the growth at infinity): simply use Lebesgue's Dominated Convergence theorem: Note first that the convergence $\|f_n-f\|\to 0$ in your $Lip_0(\mathbb R^d)$ space immediately implies pointwise a.e. convergence, $$ f_n(x)u(x)\to f(x)u(x) \qquad a.e. $$ In order to apply the DCT we only need a dominating $L^1$ bound. For this note that the Lipschitz norm controls the growth at infinity, hence $ |f_n(x)|\leq \|f_n\|\,|x|\leq 2 \|f\|\, |x| \qquad \forall x$ uniformaly in $n$. In particular given your assumptions on $u$ we get $$ |f_n(x)\, u(x)|\leq 2 \|f\|\, |x|\, u(x)\in L^1 $$ and the result follows.