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can any subgroup of the unitary group of full matrix alg $M_n(\mathbb{C})$ be approximated on finite sets by a finite subgroup?

is this true for amenable groups?

can any subgroup of the unitary group of full matrix alg $M_d(\mathbb{C})$ be approximated on finite sets by a finite subgroup?

i.e. is the following True or false? Let $n, d$ be positive integers and let $u_1,..., u_n$ be in the unitary group $U_d=U (M_d(\mathbb{C}))$ of $M_d(\mathbb{C})$. Then for every $\epsilon > 0$ there are $v_1, ..., v_n$ in $U_d$ such that $\| u_k - v_k \| < \epsilon$ for $k = 1, .., n$ and such that the subgroup of $U_d$ that $v_1, ..., v_n$ generate is finite.