I am aware that there's another post regarding this.
But I have a different proof that I would like to verify. My proof involves using existence of local section. In the textbook Introduction to smooth manifold by Jack Lee. Proposition 4.26.
I aim to prove that every points in $ \mathbb{C}^{n+1}$ is contained in the image of a smooth local section $\sigma : \mathbb{CP}^n \rightarrow \mathbb{C}^{n+1}$
Let $x$ be a point in $\mathbb{C}^{n+1}$ such that $x \in U_i$ where $U_1 = \{x \in \mathbb{C}^{n+1}$ : $ x_1 \neq 0 \}$. Then recall that $\pi(U_1)$ is an open neighborhood of $\mathbb{CP}^n$ that is locally euclidean, with parametrization $[1,x_2,x_3,.....x_{n+1}] \rightarrow (x_2,.....x_{n+1})$.
Define $\sigma: \mathbb{CP}^{n} \rightarrow \mathbb{C}^{n+1}$ where $\sigma([x_1,.....x_{n+1}]) = (z_1, z_1x_1, ......z_1x_n)$ and $z_1$ is the first coordinate of $x$. Then I think it is clear that $\sigma$ is smooth, and $\pi \circ \sigma ([x_1,.....x_{n}]) = [\frac{z_1}{z_1}, \frac{z_1x_1}{z_1}.......\frac{z_1x_{n}}{z_1}] = [1, x_1, ......,x_n]$, so $\pi \circ \sigma = id$ and $x = \sigma[1,x_1,.......,x_n]$, so $x$ is in the image of $\sigma$.
So by theorem 4.26 in the "introduction to smooth manifold" by Jack Lee, that is, a map $\pi:M \rightarrow N$ is a smooth submersion if and only if every point of $M$ is in the image of a smooth local section of $\pi$. The said quotient map is a submersion.