I'm either confused with the definition of symplectic forms / immersions or the way exercise 5.2-3 in Marsden's 'Introduction to Mechanics and Symmetry' was stated.
It reads as follows
Exercise 5.2-3. Show that any canonical map between finite-dimensional symplectic manifolds is an immersion.
My solution attempt
Let $(\mathcal P_1, \Omega_1), (\mathcal P_2, \Omega_2)$ be symplectic manifolds and $\varphi: \mathcal P_1 \to \mathcal P_2$ a canonical transformation. Let $p \in \mathcal P_1$ and $u \in \mathcal T_p \mathcal P_1 \setminus \{ 0 \}.$ Since $\Omega_1$ is non-degenerate $\exists v \in \mathcal T_p \mathcal P_1$ such that $$0 \neq (\Omega_1)_p(u, v) = (\varphi^* \Omega_2)_p(u, v) = {(\Omega_2)}_{\varphi(p)}(T_p \varphi \, u, (T_p \varphi \, v).$$ This implies $\mathrm{ker} (T_p \varphi) = \{0\}$ and since $p$ and $u$ were chosen arbitrarily $\varphi$ is an immersion. $\square$
As you notice the proof went through without any reference to the finite-dimensionality of the involved manifolds. I'm confused as I don't see why this assumption would be necessary. Did I make a mistake? Or is the exercise just stated in a misleading way?