I am trying to show that:
If $f: X \to Y$ is a non-constant holomorphic map between compact Riemann surfaces, of degree $1$, then $f$ is an immersion.
I tried proving this by contradiction. If $f$ is not an immersion, then there is some point $x$ such that $df_x: T_x(X) \to T_{f(x)}Y$ is not injective. But as $X$ is 1-dimensional, so is $T_x(X)$, and therefore $df_x$ is the zero map.
Does this imply that $f$ would be constant around $x$ which gives us a contradictory, or am I on the wrong path here?