  We introduce the notions of almost Lipschitz embeddability and nearly
isometric embeddability. We prove that for $p\in [1,\infty]$, every proper
subset of $L_p$ is almost Lipschitzly embeddable into a Banach space $X$ if and
only if $X$ contains uniformly the $\ell_p^n$'s. We also sharpen a result of N.
Kalton by showing that every stable metric space is nearly isometrically
embeddable in the class of reflexive Banach spaces.
