Tight embeddability of proper and stable metric spaces

We introduce the notions of almost Lipschitz embeddability and nearly isometric embeddability. We prove that for $p\in [1,\mathrm{\infty }]$, every proper subset of $L_p$ is almost Lipschitzly embeddable into a Banach space $X$ if and only if $X$ contains uniformly the ${\mathrm{\ell }}_p^n$’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.