I must not lie I am feeling very shaky in this subject, and I study mathematics as a hobby, so I have no way to check if my reasoning is correct. The statement of the question is exactly as in the title, "Prove that every $C^1$ submersion is an open mapping".
My idea of proof: A previous lemma from the book states that, for $f: U \to \mathbb{R}^{n}, U \subset\mathbb{R}^{m}, m \geq n, a \in U, U$ open, if $f'(a)$ is surjective, then every open ball V containing a in U contains points $x$ s.t. $|f(x)|>|f(a)|$ and, if $|f(a)| > 0$, points $y$ s.t. $|f(y)|<|f(a)|$. Because f is a submersion and hence $f'(a)$ surjective in every point a of U, the set $W = f(V)$ contains at least these $f(x)$ and $f(y)$. Because f is continuous and $|f(x)| > |f(y)|$, it follows that the set $K=\{k\in W: |f(x)| < |f(k)| < |f(y)|\}$, which is open and contained in W, contains f(a). Because the choice of a was random, we conclude $f(W)$ is open.
Are there any mistakes in my reasoning? I believe so, since I did not use the supposition of $C^1$ anywhere.