I am reading the proof of Sard's theorem from John Lee's Introduction to Smooth Manifolds. In the proof, we have $F:U \subset \mathbb{R}^m \to \mathbb{R}^n$ a smooth map and denote $C \subset U$ to be the set of critical points of $F$. Then he defines a decreasing sequence of subsets $C \supseteq C_1 \supseteq C_2 \supseteq \dots$ as follows: $$C_k = \{x\in C:\text{for } 1\le i \le k, \text{all ith order partial derivatives of $F$ vanish at x} \}.$$
Why is $C_{k+1}\subset C_k$? Isn't it possible to have a function whose $k+1$th derivative is zero but $k$th is not?