Problem:

Let A and B be two events, show that:

P(A) + P(B) -1 $\displaystyle $\leq$$ P(A U B) $\displaystyle $\leq$$ P(A)+P(B)

Questions/Attempts:

Am I supposed to assume that A and B are the only events in the sample space, and that therefore P(A)+P(B)=1?

If I assume that, and begin the proof with the first axiom of probability, I get this far:

1. 0 $\displaystyle $\leq$$ P(A) $\displaystyle $\leq$$ 1

2. P(A)+P(B)=1 (Since P(S)=1, where P(S) is the sample space)

so

3. 0 $\displaystyle $\leq$$ P(A) $\displaystyle $\leq$$ P(A) + P(B)

There's a couple things I can do from here but it seems to make things more complicated than they need to be.

Thanks for any help in advance!