10. Show that

$$P(\bigcup_{i=1}^{n} E_i) ≤ \sum_{i=1}^{n} P(E_i)$$

This is known as Boole's inequality.

Hint: Either use Equation (1.2) and mathematical induction, or else show that

$\bigcup_{i=1}^{n} E_i = \bigcup_{i=1}^{n} F_i$, where F₁ = E₁, F_i = E_i $\bigcap_{j=1}^{i-1} E_j^{c}$, and use property

(iii) of a probability.