  It is known that if finite subsets of a locally finite metric space $M$ admit
$C$-bilipschitz embeddings into $\ell_p$ $(1\le p\le \infty)$, then for every
$\epsilon>0$, the space $M$ admits a $(C+\epsilon)$-bilipschitz embedding into
$\ell_p$. The goal of this paper is to show that for $p\ne 2,\infty$ this
result is sharp in the sense that $\epsilon$ cannot be dropped out of its
statement.
