Upper bound on the independence number of a connected graph

Let $G$ be a finite simple connected graph on at least two vertices such that its complement $\overline{G}$ is also connected. For each vertex $v$ of $G$, let $l(v)$ denote the independence number of the subgraph induced by the neighborhood of $v$ in $G$. Let $\lambda = \max_{v \in V(G)} l(v)$. Let $\mathrm{diam}(\overline{G})$ denote the diameter of the complement graph $\overline{G}$, i.e., the maximum over all pairs of vertices of the length of a shortest path between them in $\overline{G}$. Prove that the independence number $\alpha(G)$ satisfies

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