Boundary of the sum of an Infinite horizon discounted model The sum of an infinite horizon discounted model is given as follows:
$$R_t = r_{t+1} + \gamma r_{t+2} + \gamma^2r_{t+3} + ... = \sum_{k=0}^\infty \gamma^kr_{t+k+1}.$$
As can be seen, the sum is bounded, but how can I show that there exists a real number $S \in \mathbb{R}$ and $ S < \infty$ such that $R_t < \infty$?
Remark: $r_t < \infty$ for all $t$
My approach is:
For any real parameter $\gamma \in (0,1)$ follows:
$$\lim_{k\to\infty} \sum_{k=0}^k\gamma^k = \frac{1}{1-\gamma}.$$ 
by simply adding the reward parameter $r_t$ to it, the boundary is given as follows:
$$R_t= \lim_{k\to\infty}\sum_{k=0}^k\left[\gamma^kr_t\right] = \frac{r_t}{1-\gamma}.$$
If I am not mistaken, this should represent the boundary. The value of $R_t$ is now limited by the real number given above.
Is this even the right approach? Any help would help me to understand this totally new topic. Thanks.
 A: We assume that the rewards $r_t$ are bounded and that $0\leq \gamma < 1$ (excluding $\gamma=1$ is important or we cannot guarantee boundedness of the return $R_t$). If the rewards are bounded we can find a number $r_0$ such that $|r_t|\leq r_0$. This inequality is a consequence of the boundedness of the reward $r_t$.
We have
$$|R_t| = |r_{t+1} + \gamma r_{t+2} + \gamma^2r_{t+3} + ...| = |\sum_{k=0}^\infty \gamma^kr_{t+k+1}|\leq \sum_{k=0}^\infty |\gamma^kr_{t+k+1}|= \sum_{k=0}^\infty |\gamma^k||r_{t+k+1}|$$
$$\leq \sum_{k=0}^\infty |\gamma^k|r_0=r_0\sum_{k=0}^\infty |\gamma|^k=r_0\dfrac{1}{1-|\gamma|}.$$
At the last step I used the formula for the infinite geometric series. Hence, we have found that 
$$|R_t|\leq r_o\dfrac{1}{1-|\gamma|} \implies R_t<\infty.$$ 
Instead of using $|r_t|\leq r_0$, we could also use the same reasoning by using $r_\text{min}\leq r_t \leq r_\text{max}$ (this follows from the boundedness of $r_t$) to conclude 
$$r_\text{min}\dfrac{1}{1-|\gamma|}\leq R_t \leq r_\text{max}\dfrac{1}{1-|\gamma|}.$$
