In the definition of a probability measure, for \( i \in \left \lbrace 1 , \ldots , n \right \rbrace \) we set \( E_{i} = A_{i} \) and then for any \( j > n \) we set \( E_{j} = \emptyset \), then we can see that \( E_{1} , E_{2} , \ldots \) is pairwise disjoint, and thus
\( P \left ( A_{1} \cup A_{2} \cup \ldots \cup A_{n} \right ) \) 
\( = \) 
\( P \left ( A_{1} \cup A_{2} \cup \ldots \cup A_{n} \cup \emptyset \cup \emptyset \cup \ldots \right ) \) 

\( = \) 
\( P \left ( \bigcup_{i = 1}^{\infty} E_{i} \right ) \) 

\( = \) 
\( \sum_{i = 1}^{\infty} P \left ( E_{i} \right ) \) 

\( = \) 
\( \sum_{i = 1}^{n} P \left ( E_{i} \right ) + P \left ( \emptyset \right ) + P \left ( \emptyset \right ) + P \left ( \emptyset \right ) + \ldots \) 

\( = \) 
\( \sum_{i = 1}^{n} P \left ( E_{i} \right ) + 0 + 0 + 0 + \ldots \) 

\( = \) 
\( \sum_{i = 1}^{n} P \left ( A_{i} \right ) \) 