Let \(A_1,\,A_2,\,A_3,\,...\subseteq S\) be an infinite sequence of possibly overlapping events in some probability space. Based on this arbitrary sequence, define a new sequence of events that starts with \(B_1 = A_1\) and then has \(B_i=A_i\cap \left(\bigcup_{j=1}^{i-1}A_j\right)^c\) for all \(i>1\).

1. Show that the \(B_i\) are pairwise disjoint.

2. Show that

\[ \bigcup_{i=1}^\infty A_i=\bigcup_{i=1}^\infty B_i. \]

3. Use the previous parts to show that

\[ P\left(\bigcup_{i=1}^\infty A_i\right)\leq\sum\limits_{i=1}^\infty P(A_i). \]