$L$ is a finite dimensional semisimple Lie algebra over a field $F$ with $F=\overline{F}$ and char$F=0$. The simple roots are $\Delta = \{\alpha_1, \ldots, \alpha_r\}$.
I want to show that there are at most $d^r$ finite dimensional simple $L$-modules $V$ with $dimV \leq d$.
I was given the following hint: Let $V$ be a finite dimensional simple $L$-module with highest weight $\lambda$ and $\alpha \in \Delta$, use the dimensions of sl$(\alpha)$-submodules to show $\langle\lambda,\alpha\rangle < dimV$.
I'm having a hard time piecing together what I know about $L$-modules to prove this, and I can't even see clearly how $\langle\lambda,\alpha\rangle < dimV$ is helpful. A hint on the strategy would be greatly appreciated, I've been banging my head against the wall for way too long with this proof.