Maximal degree of a map of surfaces

Given closed possibly nonorientable surfaces $M,N$, we prove that if a map $f:M\to N$ has degree $d>0$, then $\chi(M)\le d\cdot\chi(N)$. We give all necessary comments on the definition and properties of geometric degree, which can be defined for any map. Our proof is based on the Kneser-Edmonds factorization theorem, simple natural proof of which is also presented.