The whole turns on a simple application of the triangle inequality.
ok so how do you know that abs(an - A) + abs(bn - B) is less than epsilon? you know what i'm saying? we know that that expression and that epsilon are both greater than or equal to abs((an + bn) - (A+B)). how do we know that abs(an - A) + abs(bn - B) is less than epsilon?
you said that abs(an - A) is less than epsilon/2, but we don't know this, right?
it's like the triangle inequality doesn't work because it's reversed.