# Show that all irreducible representations of a finite group are finite dimensional.

Show that all irreducible representations of a finite group are finite dimensional.
We know for any representation $V$ of a finite group $G$, there is a decomposition $V=V_1 \oplus V_2 \oplus.....\oplus V_n$ where $V_i$ s are all irreducibles. Now if we make $V$ irreducibles then just one $V=V_1$(say) will be there. But how can we show $V$ is finite dimensional?

Any help will be apreciated..

Pick $v\in V$ nonzero and consider $\Bbb C[G]v$, the $\Bbb C$-span of $v$‘s orbit under $G$.