  Assume that $g(|\xi|^2)$, $\xi\in\mathbb{R}^k$, is for every dimension
$k\in\mathbb{N}$ the characteristic function of an infinitely divisible random
variable $X^k$. By a classical result of Schoenberg $f:=-\log g$ is a Bernstein
function. We give a simple probabilistic proof of this result starting from the
observation that $X^k = X_1^k$ can be embedded into a L\'evy process
$(X_t^k)_{t\geq 0}$ and that Schoenberg's theorem says that $(X_t^k)_{t\geq 0}$
is subordinate to a Brownian motion. A key ingredient of our proof are concrete
formulae which connect the transition densities, resp., L\'evy measures of
subordinated Brownian motions across different dimensions. As a by-product of
our proof we obtain a gradient estimate for the transition semigroup of a
subordinated Brownian motion.
