Infinite-dimensional general linear groups are groups of universally finite width

Recently George Bergman proved that the symmetric group of an infinite set possesses the following property which we call by the {\it universality of finite width}: given any generating set $X$ of the symmetric group of an infinite set $\Omega,$ there is a uniform bound $k \in \N$ such that any permutation $\sigma \in \text{Sym}(\Omega)$ is a product of at most $k$ elements of $X \cup X^{-1},$ or, in other words, $\text{Sym}(\Omega)=(X^{\pm 1})^k.$ Bergman also formulated a sort of general conjecture stating that `the automorphism groups of structures that can be put together out of many isomorphic copies of themselves' might be groups of universally finite width and particularly mentioned, in this respect, infinite-dimensional linear groups. In this note we confirm Bergman's conjecture for infinite-dimensional linear groups over division rings.