On the density of vector fields with only nondegenerate zeroes

Question

Suppose we have a manifold $X$ embedded in $\mathbb{R}^n$. Define the vector field $v_u(p) : X \rightarrow TX$ by taking the point $u \in \mathbb{R}^n$ to its natural (orthogonal) projection onto $TX_p$. How can we show that the set of $u \in \mathbb{R}^n$ such that every zero of $v_u$ is nondegenerate is dense in $\mathbb{R}^n$?