Borel-Cantelli Lemma Proof. !enter image description here **I struggle to understand the transition between the steps in the red box.** Especially why the limit there, i get the intuition behind the limit, its because $A_n$ is a decreasing set ie $ A_1\supseteq \ A_2 \supseteq\ A_3\supseteq ...$ but is there any rigour to it? The union and intersection parts make sense though! Relevant thm 1.25c) post is: $A_n$ is an increasing set. Then what is $\ A_1\setminus A_0$ if $A_0=\emptyset$ .

The sequence $(A_n)_{n \in \mathbb{N}}$ is not necessarily decreasing. However, if we define

$$B_n := \bigcup_{k \geq n} A_k$$

then $B_n$ is decreasing since

$$B_{n+1} = \bigcup_{k \geq n+1} A_k \subset \bigcup_{k \geq n} A_k = B_n.$$

Since $B_n \downarrow B$ for $B:= \bigcap_{n \geq 1} B_k$, the continuity of the measure from above shows

$$\begin{align*} \mathbb{P} \left( \bigcap_{n \geq 1} \bigcup_{k \geq n} A_k \right) &= \mathbb{P}(B) \\\ &= \lim_{n \to \infty} \mathbb{P}(B_n) \\\ &= \lim_{n \to \infty} \mathbb{P} \left( \bigcup_{k \geq n} A_k \right). \end{align*}$$