Simple proof that the arithmetic genus is non-negative

Mumford (Complex Projective Varieties, section 7) has the following, reasonably simple proof. Let $d$ be the degree of $\mathrm{C}$, $m$ big enough such that $h_{\mathrm{C}}(m)=\mathrm{dim}_{\mathbf{C}} (\mathbf{C}[\mathrm{T}_0,\ldots,\mathrm{T}_n]/\mathrm{I}(\mathrm{C}))_m$ and $md/2>p_a$. Embed $\mathrm{C}$ into $\mathbf{P}^N$ by the degree $m$ Veronese embedding; let $\mathrm{L}\subset\mathbf{P}^N$ be a linear space containing $\nu_m(\mathrm{C})$ such that $\nu_m(\mathrm{C})$ is nondegenerate in $\mathrm{L}$. Then $\dim(\mathrm{L})=h_\mathrm{C}(m)=md+1-p_a$, in other words $\mathrm{L}\simeq\mathbf{P}^{md-p_a}$, and the degree of $\nu_m(\mathrm{C})$ is $$md=\mathrm{deg}(\nu_m(\mathrm{C}))\geqslant\mathrm{codim}(\nu_m(\mathrm{C}))+1\geqslant md-p_a$$ since $\nu_m(\mathrm{C})$ is nondegenerate in $\mathrm{L}$.