Probability That None of Independent Events Occurs

The problem says:

Let $\displaystyle A_1, A_2, A_3,...,A_n$ be independent events. Show that the probability that none of the $\displaystyle A_i's$ occur is less than or equal to exp(-$\displaystyle \Sigma_{i=1}^{n}P(A_i))$.

I'm confused at where the inequality and the exponent are coming from.

Here is what i've done.

I figured that the P(none happening) = 1 - P(one happening)

P(one happening) = $\displaystyle \Sigma_{k=1}^{n}P(A_k)$.

Then, P(none) = 1 - $\displaystyle \Sigma_{k=1}^{n}P(A_k)$.

I think this is right, but where to go from here, I'm not sure.

Thank you!