In the proof of Sard's Theorem presented in John M. Lee's Introduction to Smooth Manifolds i.e. when trying to show that the set of critical values of a smooth map $F : M \rightarrow N$ between smooth manifolds has measure zero, we identify $M$ with a subset $U \subseteq \mathbb{R}^n$ and let $C \subseteq U$ denote the set of critical points of $M$. We also define a decreasing sequence of sets $C \supseteq C_1 \supseteq C_2 \cdots $ where $C_k$ is the set of points $x \in C$ where all i-th order partial derivatives of $F$ vanish.
It's noted that $C$ and each $C_k$ are closed in $U$, by continuity. Why is this true?