Proving inf A + inf B ≤ inf(A+B): Is it True?

[gauss mouse]

Hi, I'm just wondering this:
If A and B are subsets of the real numbers, is
inf A + inf B = inf(A+B)?

I've proved that
inf A + inf B <or= inf(A+B)?

I've tried the rest of the proof but it won't work. Is it even true?

 

[mathman]

Let AI = inf(A), BI = inf(B) for every ε > 0, there exists a (in A) and b (in B) such that
a < AI + ε and b < BI + ε, a + b < AI + BI + 2ε, inf(A+B) < AI + BI + 2ε .

Since ε is arbitrarily small, inf(A+B) ≤ inf(A) + inf(B).

 

[gauss mouse]

Let AI = inf(A), BI = inf(B) for every ε > 0, there exists a (in A) and b (in B) such that
a < AI + ε and b < BI + ε, a + b < AI + BI + 2ε, inf(A+B) < AI + BI + 2ε .

Since ε is arbitrarily small, inf(A+B) ≤ inf(A) + inf(B).

Thanks. I got it in the end but your way is much prettier.