Consider the rational projection map from a point $p$, $\pi_p:\mathbb{P}^n\longrightarrow \mathbb{P}^{n-1}$ given by sending $q\neq p$ to the intersection of the line containing $q,p$ with $\mathbb{P}^{n-1}$, and sending $p$ to $\infty$. Consider the image $\pi_p(C)$ for some curve $C\subset \mathbb{P}^n$.
I am trying to determine the bound $\deg(\pi_p(C))\leq \deg(C)-1$.
How does one go about this?
Edit: the map $\pi_p$ is not defined at $p$ (see @Kreiser's comment below)
Let's assume $n\ge 3$, so the target is not $\mathbb P^1$ and $\pi_p|_C:C\to \pi_p(C)$ is birational.
Choose a general hyperplane $H\subseteq \mathbb P^{n-1}$ that intersects $\pi_p(C)$ transversely at $m$ points. So $$\deg(\pi_p(C))=\#(H\cap \pi_p(C))=m.$$
Now, the cone of $H$ and $p$ defines a hyperplane $\tilde{H}$ of $\mathbb P^{n}$, which intersects $C$ at the preimages of those $m$ points plus a point $p$. By generality, all intersections are simple, so
$$\deg(C)=\#(\tilde{H}\cap C)=1+\deg(\pi_p(C)).$$