Let $x\in M_{\mathbb{R}}$ such that $H^{\vee}(x)\in\operatorname{stab}(g)$. By the definition of the stability set, $\sup_{v\in Q_{\mathbb{R}}}(g(v)-\langle H^{\vee}(x),v\rangle)<\infty$. Thus, for any $u\in N_{\mathbb{R}}$,

	$\displaystyle\sup_{v\in Q_{\mathbb{R}}}(g(v)-\langle H^{\vee}(x),v\rangle)$	$\displaystyle=\sup_{v\in Q_{\mathbb{R}}}(g(v)-\langle x,H(v)\rangle)$
		$\displaystyle\geq\sup_{v\in H^{-1}(u)}(g(v)-\langle x,H(v)\rangle)=\sup_{v\in H^{-1}(u)}g(v)-\langle x,u\rangle$

and so $\sup_{v\in H^{-1}(u)}g(v)$ is bounded above, as stated. ∎