Hartshorne Exercise IV.1.6.
Let $X$ be a curve of genus $g$. Show that there is a finite morphism $f:X\ra\PP{k}{1}$ of degree $\le g+1$. (Recall that the degree of a finite morphism of curves $f:X\ra Y$ is defined as the degree of the field extension $[\Bbb K(X):\Bbb K(Y)]$ (II, §6).)
First note that if a line bundle $\mathcal{L}$ on a (smooth, proper) curve $X$ has at least two linearly independent global sections, then this linear system induces a dominant morphism of curves $f:X\ra \PP{k}{1}$ and $\deg f=\deg \mathcal{L}$. Indeed, we already know that a non-constant birational map $X\dashrightarrow\PP{k}{1}$ (e.g. one induced by 2 sections of $\mathcal{L}$) extends to a finite, surjective morphism $X\ra \PP{k}{1}$. We have $\mathcal{L}\cong f^*\OO(1)\implies \deg \mathcal{L}=\deg f$.
By RR, for any point $P\in X$
$$h^0(\OO_X(nP))=h^1(\OO_X(nP))+n-g+1.$$
Choosing $n=g+1-h^1(\OO_X(nP))$ if $h^1(\OO_X(nP))<g+1$ and $n=1$ if $h^1(\OO_X(nP))\ge g+1$ yields a complete linear system of dimension $\ge 2$. Choosing two linearly independent global sections gives the situation above.
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