This is an excerpt taken from Probability with Martingales by Williams.
The framework is probability theory.
Why is the equation being discussed true if condition $\{ n \ge m \}$ is replaced by condition $\{ r \ge n \ge m \}$?
And what is meant then saying that the limit as $r \uparrow \infty$ is justified by the monotonicity of the two sides? Is he taking the limit?
In general I do not understand what is happening in that part of the proof :)
This is a notation issue: $$\Pi_{n \geq m} (1-p_n) = \lim_{r \to \infty} \Pi_{r \geq n \geq m} (1-p_n). $$ This is the analog of $\sum_{n = 1}^\infty a_n = \lim_{r \to \infty} \sum_{n=1}^r a_n$. The infinite product is the limit (when it exists) of the partial products
You might see this as a consequence of the limit in the sets under consideration: $$\cap_{n \geq m} A_n^c = \lim_{r \to \infty} \cap_{r \geq n \geq m} A_n^c $$ and
$$P(\cap_{n \geq m} A_n^c) = P(\lim_{r \to \infty} \cap_{r \geq n \geq m} A_n^c) = \lim_r P(\cap_{r \geq n \geq m} A_n^c) \overset{\text{ind.}}{=} \lim_r \Pi_{r \geq n \geq m} P(A_n^c) = \lim_r \Pi_{r \geq n \geq m}(1-p_n) $$