I would like to know why:
$\left| \sum\limits_{t=0}^\infty \delta^tr(s_t,a_t)\right|\le \sum\limits_{t=0}^\infty \delta^t|r(s_t,a_t)|$
where:
$\delta \in (0,1)$ and is a $r(s_t,a_t)$ is a real valued function and is bounded.
My text states that the above is a consequence of the Cauchy-Schwarz inequality, but it is not immediately clear to me why it is true. So, I tried to prove it by starting from the Cauchy-Schwarz inequality:
$\begin{align} \left(\sum\limits_{i=0}^n a_ib_i\right)^2 \le \left(\sum\limits_{i=0}^n a_i^2\right) \left(\sum\limits_{i=0}^nb_i^2\right) \end{align}$
Taking square roots on both sides, I can get: $$ \left| \sum\limits_{i=0}^n a_ib_i \right| \le \left(\sum\limits_{i=0}^n a_i^2\right)^\frac{1}{2} \left(\sum\limits_{i=0}^nb_i^2\right)^\frac{1}{2} \le \left(\sum\limits_{i=0}^n a_i^2\right) \left(\sum\limits_{i=0}^nb_i^2\right)$$
Now, I don't know what else I can do to get $\left(\sum\limits_{i=0}^n a_i|b_i|\right)$