Corollary 2

Assume that G is semi-simple; let $d$ be an integer. The set of classes of representations of G of dimension $\leq d$ is finite.

Cor. $1b)$ implies that the set $X_d$ of elements $\lambda$ of $X_{++}$ such that $d(\lambda )\leq d$ is finite. For all $\lambda$ in $X_d$, let $V_{\lambda}$ be an irreducible representation of highest weight $\lambda$; every representation of dimension $\leq d$ is isomorphic to a direct sum $\bigoplus_{\lambda\in X_d}V^{n_{\lambda}}_{\lambda}$, with $n_{\lambda}\leq d$, hence the corollary.