  The main purpose of the paper is to prove the following results:
  Let $A$ be a locally finite metric space whose finite subsets admit uniformly
bilipschitz embeddings into a Banach space $X$. Then $A$ admits a bilipschitz
embedding into $X$.
  Let $A$ be a locally finite metric space whose finite subsets admit uniformly
coarse embeddings into a Banach space $X$. Then $A$ admits a coarse embedding
into $X$.
  These results generalize previously known results of the same type due to
Brown-Guentner (2005), Baudier (2007), Baudier-Lancien (2008), and the author
(2006, 2009).
  One of the main steps in the proof is: each locally finite subset of an
ultraproduct $X^\mathcal{U}$ admits a bilipschitz embedding into $X$. We
explain how this result can be used to prove analogues of the main results for
other classes of embeddings.
