Suppose that $M$ and $N$ are closed connected oriented surfaces. It is well-known that **if $f \colon M \to N$ has degree $d > 0$, then $\chi(M) \le d \cdot \chi(N)$.**

What is an elementary proof of this?

I found references to some H.Kneser's articles (1928, 1929, 1930). However, I am not sure that these give a complete proof. Also they are written in German with obsolete notation. Is there a survey on maps of surface where I can find the proof in English?

Michael Albanese suggests that we use the Gromov norm. This path seems complicated (to me). Is it possible that the original proof (before Gromov) is more elementary?

For example, if $\chi(M) > \chi(N)$, then one can easily prove that $d$ is zero, using the quadratic form on $H^1(N ; \mathbb{Z})$. Can we use this approach, or something similar, to get a precise estimate on $d$ in general?