Does this proof in Puterman's MDP book make sense?

I'm reading Chapter 6 of Puterman's MDP :Discrete Stocastic Dynamic Prorgamming. In the book, Bellman operator $$\mathscr{L}$$ is given as $$\begin{equation} (\mathscr{L}v)(s) = \sup_{d}\{r_d(s) + \lambda P_d v(s)\},\quad s\in S, v\in\mathbb{R}^S, \end{equation}$$ where $$r_d(s)$$ is expected immdiate return $$r(s,a)$$ following the policy $$d$$ given the state $$s$$, $$P_d$$ is the trasition matrix corresponding to $$d$$, and $$0\leq\lambda<1$$ is the discounted rate. The supremum is taken over all possible policies d.

Then, (b) of the theorem 6.2.2 states that if $$v\leq\mathscr{L}v$$, then $$v\leq v^{*}_\lambda$$, where $$v^{*}_\lambda$$ is defined by $$\begin{equation*} v^{*}_\lambda = \sup_{\pi}v^{\pi}_\lambda(s). \end{equation*}$$ Here, $$v^{\pi}_\lambda(s)$$ is the value following policy $$\pi$$ with the initial state $$s$$.

The proof proceeds as follows: if $$v\leq \mathscr{L}v$$, then for any $$\epsilon>0$$, there exists some $$d$$ such that $$v\leq r_d + \lambda P_d v + \epsilon e$$, where we denote $$(1, 1,\cdots)^{\top}$$ by $$e$$, and the inequality applies for each $$s$$. Then, after playing with some definitions and inequalities, the proof ends.

What makes me confused is, however, taking such $$d$$ depends on each state $$s$$, so taking such a universal $$d$$ seems inappropriate. My question is

Am I missing something, or does the proof have a flaw?