Let $X\subset \mathbb{P}^n$ be a reduced closed subscheme. For a general hyperplane $H$, $X\cap H$ is again reduced (and of dimension one less). Is there an easy proof of this result?

Algebraically, we take an algebraically closed field $k$, an homogeneous radical ideal $I\subset k[X_0,\ldots,X_n]$ and ask that for $a_0,\ldots,a_n\in k$ general, the ideal generated by $I$ and $a_0X_0+\cdots +a_nX_n$ is again radical (and of dimension one less).

The result follows from Bertini theorem if $X$ is smooth, but I do not assume it here, and would like to have a simple proof of this.