  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.
