Why is $\{x\in M: F(x)=G(x), DF_x=DG_x\}$ closed? Let $F$ and $G$ be smooth maps between smooth manifolds $M$ and $N$. Denote their differentials at $x\in M$ by $DF_x$ and $DG_x$. Why is the set $\{x\in M:F(x)=G(x), DF_x=DG_x\}$ closed in $M$?
I know of the result that if $f$ and $g$ are continuous maps between topological spaces $X$ and $Y$ with $Y$ being Hausdorff then $\{x\in X:f(x)=g(x)\}$ is closed in $X$. Unfortunately it doesn't seem like I can directly apply this result since I have $DF_x$ and $DG_x$ which are now maps between tangent spaces.

Take a sequence $x_n\to x$ and note that if
$$
F(x_n)=G(x_n)\quad\text{and}\quad DF_{x_n}=DG_{x_n}
$$
for all $n$, then
$$
F(x)=G(x)\quad\text{and}\quad DF_{x}=DG_{x}.
$$
