Let $X$ be a projective variety in $\mathbb{P}^n$, and let $x\in X$ be a regular point. How can prove that there exists a hyperplane $H$ that contains $x$ and intersect $X$ transversally?
I think one should try to think that the space of hyperplanes through $x$ that do not satisfy this condition is a proper closed subset of the space of hyperplane through $x$ (identified with $\mathbb{P}^{n-1}$). But how to proceed further?