Discounted rewards

Discounted rewards is a way to evaluate a Markov decision process with infinite steps. Let $r_i$ for $i \in \mathbb{N}$ be a sequence of rewards for this decision process. Let $\gamma \in [0,1)$ be some constant. Then we set the value of these choices as

$$ > V(r) = \sum_{i \in \mathbb{N}} \gamma^i r_i. > $$

If all $r_i$ are bounded by some constant $R_{max}$ then we have

$$ \sum_{i=0}^{\infty} \gamma^i r_i \leq \frac{R_{max}}{(1 - \gamma)}$$

and this is finite.