Let $S$ be a maximum independent set, so $|S| = \alpha(G)$. Since no edge has both endpoints in $S$, every edge is incident to at most one vertex of $S$, and hence $\sum_{v \in S} d(v) \le m$. The $\alpha(G)$ smallest degrees sum to at most the degrees of any $\alpha(G)$ vertices, in particular of those in $S$. Therefore $\sum_{i=1}^{\alpha(G)} d_i \le m$, and $\alpha(G) \le a(G)$ by the maximality in the definition of $a(G)$.