On the algebraic structure of the unitary group

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