I have two functions $f,g: \mathbb{R} \rightarrow \mathbb{R}$ which are continuous. Now in a proof one step that is not further explained says that the set
$M = \{x \in \mathbb{R}| f(x) \leq g(f(x))\}$ is closed.
I thought about it but could not find a short formal argument and I fear the answer is very trivial because the book explains all others steps very detailed. I noticed that if you instead say $f(x) < g(f(x))$ it is not true anymore because you could set $f(x)=\frac{1}{|x|+1}$ and $g(x)=1$ getting a contradiction for a sequence with $x_n \rightarrow 0$. Thank you in advance.