$X$ is a finite disjoint union of finite orbits: $X = \cup_{i}Gx_{i}$ We can separate this union by considerating first only the orbits of 1 element and then the rest: $X = \cup_{j=1}^{l}\{x_{i_{j}}\} \cup \cup_{k=1}^{s}Gx_{i_{k}}=G_{X} \cup_{k=1}^{s}Gx_{i_{k}}$ Then using the orbit-stabilizer theorem, we have $\#X=\#G_{X} + \sum_{k=1}^{s}[G:G_{x_{i_{k}}}]$ where for every $k$ $[G:G_{x_{i_{k}}}]\geq 2$ because if one of them were 1, then it would be associated to an orbit of 1 element, but we counted those orbits first. Then this stabilizers are not $G$ This finishes the proof.