Stability of Tautological Bundles on Symmetric Products of Curves

We prove that, if $C$ is a smooth projective curve over the complex numbers, and $E$ is a stable vector bundle on $C$ whose slope does not lie in the interval $[-1,n-1]$, then the associated tautological bundle $E^{[n]}$ on the symmetric product $C^{(n)}$ is again stable. Also, if $E$ is semi-stable and its slope does not lie in the interval $(-1,n-1)$, then $E^{[n]}$ is semi-stable.