Uniform embeddings of bounded geometry spaces into reflexive Banach space

We show that every metric space with bounded geometry uniformly embeds into a direct sum of $l^p ({\mathbb N})$ spaces ($p$'s going off to infinity). In particular, every sequence of expanding graphs uniformly embeds into such a reflexive Banach space even though no such sequence uniformly embeds into a fixed $l^p ({\mathbb N})$ space. In the case of discrete groups we prove the analogue of a-$T$-menability - the existence of a metrically proper affine isometric action on a direct sum of $l^p ({\mathbb N})$ spaces.