An existence of non-constant meromorphic functions on an arbitrary compact Riemann surface is a non-trivial and important fact in algebraic geometry, which is used, for example, in the elementary proof of the Riemann-Roch theorem. One way how we can prove it is based on a finite dimensionality of $H^1(X,\mathscr{O}_X)$ where $X$ is a compact Riemann surface and $\mathscr{O}_X$ is a structure sheaf (i.e., the sheaf of holomorphic functions).