Let us work over an algebraically closed field $K$. If $X\subset \mathbb{P}^n$ is a closed subset of dimension $r$, then there should exist a linear subspace $L\subset \mathbb{P}^n$ of dimension $n-r$ such that $X\cap L=\{x_1,\ldots,x_k\}$ is finite, and such that for each $i\in \{1,\ldots,k\}$ the tangent spaces of $X$ and $L$ at $x_i$ intersect only at $0$ (or at $x_i$ if you look at the projective tangent spaces, embedded in $\mathbb{P}^n$).
Is there an easy proof of this existence?
Over $K=\mathbb{C}$, in Mumford, Algebraic Geometry, Complex Algebraic Geometry, Theorem 5.1, not only the existence is claimed, but also the fact that the number $k$ is constant (equal to the degree of $X$, either as a definition, or as a theorem, depending on how you define the degree). The fact that $k$ is constant is explained in a long proof, and the existence is proven by taking a linear subspace $L'\subset \mathbb{P}^n$ of dimension $n-r-1$, projecting from $L'$ and taking a point of $\mathbb{P}^r$ over which the projection is smooth. However, the fact that the projection $X\dashrightarrow \mathbb{P}^r$ is smooth on a general point of $\mathbb{P}^r$ is claimed without any argument. Why is it true? It seems to me that it is false in positive characteristic, but maybe it is true for a right choice of $L'$ ? Or using another proof?
