For any limit process it is not just about bounding the sum (or function/sequence etc.) it is about bounding its difference from the actual limit value. Here you are bounding the difference between the approximating sum and 0. I.e. just the absolute value of the approximating sum. That's why the limit is 0.
I'd recommend looking at theorem 6.7 (c).
I'd recommend looking at theorem 6.7 (c).