Show that $GL_n(\mathbb{F})$ is a finite group iff $\mathbb{F}$ has a finite number of elements
Looks great. The only adjustment I'd make is to say that $\left|GL_n(\Bbb F)\right|\le m^{n^2}.$ It's certainly true that $\left|GL_n(\Bbb F)\right|<m^{n^2},$ but you haven't justified it, and the strictness isn't necessary for finiteness. That's largely an aesthetic issue, though, not a technical one.