Question about Milnor's proof of Sard's Theorem

Question

We've just covered Sard's theorem and have just started to look at transversality in my differential geometry class and I'm trying to understand a proof of Sard's theorem (based on Milnor's proof): If $F:M^m \to N^n$ is a smooth map between smooth manifolds of dimension $m$ and $n$ respectively, then $F(C)$ has measure zero in $N$ (where $C = \{x \in M : {rank}\; dF_x < n\}$). The strategy behind the proof seems clear: write $C = (C\smallsetminus C_1)\cup (C_1 \smallsetminus C_2) \cup \cdots \cup (C_{k-1} \smallsetminus C_k) \cup C_k$ (where $C_i$ denotes the set of points in $M$ for which all partial derivatives of order $\le i$ vanish at $x$) and then show that the image under $F$ of each of the sets in the union on the r.h.s. has measure zero in $N$. But I'm a little confused on a few details and in particular I was wondering why each of the sets in the descending sequence $C \supset C_1 \supset C_2 \supset C_3 \supset \cdots$ is necessarily closed.