Schoenberg (Ann. of Math. 39 (1938), 811-841) observed that if $f(t)$ is a [completely monotonic function][1], then the radial kernel $K(x,y)=f(\|x-y\|^2)$ is positive definite on any Hilbert space. Since 
$$ f(t):=\frac{1}{1+\frac{t}{\sigma^2}} $$ is completely monotonic, the result follows.

**Added.** Schoenberg's proof relies on the [Hausdorff-Bernstein-Widder theorem][2] and the fact that the Gaussian kernel $\exp(-\|x-y\|^2)$ is positive definite. Using these two facts, the proof is immediate.


  [1]: http://mathworld.wolfram.com/CompletelyMonotonicFunction.html
  [2]: http://en.wikipedia.org/wiki/Bernstein%27s_theorem_on_monotone_functions