  In this article it is proved, that every locally compact second countable
group has a left invariant metric d, which generates the topology on G, and
which is proper, ie. every closed d-bounded set in G is compact. Moreover, we
obtain the following extension of a result due to N. Brown and E. Guentner:
Every locally compact second countable $G$ admits a proper affine action on the
reflexive and strictly convex Banach space $\bigoplus^{\infty}_{n=1} L^{2n}(G,
d\mu),$ where the direct sum is taken in the $l^2$-sense.
