If $A_1, A_2, .....A_n $ are any n events, then

$P(A_1 ∪ A_2 ∪ ....∪ A_n) ≤ P(A_1) + P(A_2) + ....+ P(A_n)$

For any two events A and B, we have

$P(A ∪ B) = P(A) + P(B) - P(A ∩ B)$

$⇒ P(A ∩ B) ≤ P(A) + P(B)$

Using principle of mathematical induction it can be easily proved that

$P\left(\bigcup\limits_{i=1}^nA_i\right) ≤ \sum\limits^{n}_{i=1} P(A_i)$