Embeddability of locally finite metric spaces into Banach spaces is finitely determined

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.