Chapter 7 - Section 7.2 - Probability Theory - Exercises - Page 467: 15

Let $X_i=E_i-\cup^{i-1}_{j=1}E_j$

Let $X_i=E_i-\cup^{i-1}_{j=1}E_j$ $X_i$'s are disjoint subsets of the sample space so $P(\cup ^n_{i=1} X_i)=\sum^n_{i=1}P(X_i)$ and $X_i\subseteq E_i$ implies $P(X_i)\leq P(E_i)$. $$ P(\cup ^n_{i=1} E_i)= P(\cup ^n_{i=1} X_i)=\sum^n_{i=1}P(X_i)\leq\sum^n_{i=1}P(E_i)$$