I have often seen the assertion that for a smooth plane curve $C$ of degree $d$ the gonality of $C$ is $d-1$ and each gonality pencil is obtained by projection from a point of $C$ onto a line.

(let me recall that the gonality of $C$ is by definition the minimal degree $d$ of divisors $D$ on $C$ with $r(D)=1$ and those divisors which attain the minimum being called gonality pencils)

Is there a relatively simple proof for this fact?