  The main result of the paper: Given any $\varepsilon>0$, every locally finite
subset of $\ell_2$ admits a $(1+\varepsilon)$-bilipschitz embedding into an
arbitrary infinite-dimensional Banach space.
  The result is based on two results which are of independent interest:
  (1) A direct sum of two finite-dimensional Euclidean spaces contains a
sub-sum of a controlled dimension which is $\varepsilon$-close to a direct sum
with respect to a $1$-unconditional basis in a two-dimensional space.
  (2) For any finite-dimensional Banach space $Y$ and its direct sum $X$ with
itself with respect to a $1$-unconditional basis in a two-dimensional space,
there exists a $(1+\varepsilon)$-bilipschitz embedding of $Y$ into $X$ which on
a small ball coincides with the identity map onto the first summand and on a
complement of a large ball coincides with the identity map onto the second
summand.
