Degree and maps between closed orientable surfaces

Degree and maps between closed orientable surfaces

Let $M_g, M_h$ be closed orientable surfaces of genus $g,h$ respectively.
If $g>h$, we know there exists a map $M_g \rightarrow M_h$ of degree 1:
just think of $M_g=M_h\#M_{g-h}$ and consider the map $M_g=M_h\#M_{g-h}
\rightarrow M_h$ that pinches $M_{g-h}$ to a point, which can be easily
seen to have degree 1.
How about the case $g<h$? I have read that any map $f:M_g \rightarrow M_h$
with $g<h$ must have zero degree (and hence be homotopic to a constant
map).
One way to see this would be to show that $f$ is non-surjective. How can
we do this? The induced map in homology $f_{\ast}:H_1(M_g)\simeq
\mathbb{Z}^{2g} \rightarrow H_1(M_h) \simeq \mathbb{Z}^{2h}$ is clearly
non-surjective, but how about $f$? Is this the right way to proceed?