We consider the unitary group \U of complex, separable,
infinite-dimensional Hilbert space as a discrete group. It is proved that,
whenever \U acts by isometries on a metric space, every orbit is bounded.
Equivalently, \U is not the union of a countable chain of proper subgroups,
and whenever \E\subseteq \U generates \U, it does so by words of a fixed
finite length
