Gonality bound from Riemann-Roch ($\mathrm{gon}(X)\le g+1$)

For a point $P$ on a smooth projective genus-$g$ curve, Riemann-Roch space $L((g+1)P)$ has dimension at least two. A nonconstant function in it has only a pole at $P$, of order at most $g+1$, and extends to a finite morphism to the projective line. The degree equals the total pole order, including inseparable degree if present. This proves the bound.