Prove Bonferroni’s inequality I have read other solutions regarding proof of Bonferroni’s inequality.
However, is my derivation correct?
Suppose that E and F are two events of a sample space S.
Conclude: P(EF) ≥ P(E) + P(F) − 1
Proof:
P(EF)= 1 - P((EF)^c)
     = 1 - P(E^c  ∪ F^c) 
     = 1 - (P(E^c) + P(F^c))
     = 1 - ((1 - P(E)) + (1 - P(F)))
     = 1 + -1 + P(E) -1 + P(F) 
     = 1 - P(E) + P(F)


You seem to assume that $E^c$ and $F^c$ are disjoint in writing 
$$ 1 - P(E^c \cup F^c)  = 1 -[P(E^c) + P(F^c)].$$ 
(Also, you don't write any inequalities in your proof. Though maybe you meant to use an inequality at precisely this step...)
A simple proof notes that in general we have, $$P(E \cap F) = P(E) + P(F) - P(E \cup F).$$ 
And this implies the result because $P(E \cup F) \leq 1$.
