  In this paper we prove finiteness principles for $C^{m}\left( \mathbb{R}^{n},
\mathbb{R}^{D}\right) $-selection, and for $C^{m-1,1}\left( \mathbb{R}^{n},
\mathbb{R}^{D}\right) $-selection, in particular providing a proof for a
conjecture of Brudyni-Shvartsman (1994) on Lipschitz selections for the case
when the domain is $X = \mathbb{R}^n$. Our results raise the hope that one can
start to understand constrained interpolation problems in which e.g. the
interpolating function $F$ is required to be nonnegative everywhere.
