Lemma (5.5.1)

Let $\mathbb{A}$ be a set and $\{ f_{n} \}$ be a sequence of maps s.t. $f_{n}:\mathbb{A}\to \mathbb{R}, \forall n\in\mathbb{N}$. Then $$\limsup_{ n \to \infty }\inf_{x\in\mathbb{A}}f_{n}(x)\le \inf_{x\in\mathbb{A}}\limsup_{ n \to \infty }f_{n}(x)$$

Lemma (5.5.2)

Let $V_{n}(x,u)\uparrow V(x,u)$ pointwise. Suppose that $V_{n}$ and $V$ are continuous in $u$ for every $x$, and $u\in\mathbb{U}(x)=\mathbb{U}$ is compact. Then, $$\lim_{ n \to \infty } \min_{u\in\mathbb{U}(x)}V_{n}(x,u)=\min_{u\in\mathbb{U}(x)}V(x,u)$$

Lemma (5.5.3)

Define $\mathbb{T}:v\mapsto \mathbb{T}(v)$ as our DCOE

1. If $v$ is a measurable $\mathbb{R}_{+}-$valued function under Measurable Selection Conditions such that $v\ge \mathbb{T}(v)$ then, $$v(x)\ge J_{\beta}(x)$$
2. Let $v\le T(v)$ and $$\lim_{ n \to \infty }\beta^{n}E_{x}^{\gamma}[v(x_{n})]=0,\forall x\in\mathbb{X},\gamma\in\Gamma_{A}$$Then $$v(x)\le J_{\beta}(x)$$

Lemma (5.5.4)

If $$v(x)=\lim_{ T \to \infty } J_{\beta}^{T}(x)$$is so that $v=\mathbb{T}(v)$ where, $$\mathbb{T}(v)(x)=c(x,f(x))+\beta E[v(x_{1})|x_{0}=x,u_{0}=f(x_{0})]$$ is such that with $\gamma=\{ f, f, \dots\}$, $$\lim_{ n \to \infty } \beta^{n}E_{x}^{\gamma}[v(x_{n})]=0$$then, $\gamma$ is optimal.