Assume that g(|ξ| ² ), ξ∈R ^k , is for every dimension k∈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 Lévy process (X _t ^k ) _t≥0 and that Schoenberg's theorem says that (X _t ^k ) _t≥0 is subordinate to a Brownian motion. Key ingredients in our proof are concrete formulae which connect the transition densities, resp., Lévy 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.