I encounter the following problem. Given a compact Riemann surface $X$, and for any point $p \in X$, prove there exists a meromorphic function $f$ such that $f$ is holomorphic on $X\setminus \{p\}$.
I try to use the fact that given points $p$, $q_1,\dots ,q_m$ there exists a meromorphic function $g$ such that $g$ is holomorphic on points $q_1, \dots, q_m$ and has a pole at $p$. But I don't know how to start, could anyone give a hint or a solution?
This follows from Riemann-Roch: https://en.wikipedia.org/wiki/Riemann%E2%80%93Roch_theorem. In the notation from wikipedia, we let $D = g[P]$, since in that case we have $l(D) = deg(D) - g + 1 = 1$, meaning that the dimension of meromorphic functions $h$ with $(h) + D \geq 0$ is positive. In particular, there is a non-zero meromorphic function $h$ with $(h) + g[P] \geq 0$, so its only pole is at $p$.