On embeddings of locally finite metric spaces into ${\mathrm{\ell }}_p$

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