As a result we have a simple result.
Theorem 1: Let be a convergent sequence of non-negative terms. Then its limits is non-negative. (This should be obvious from the link I gave you).
Theorem 2: Let for all . And are convergent sequences. Then .
Proof: Define a sequence . Now this means that . And furthermore is convergent since it is a difference of two convergent sequences. Thus, . Thus, because the limit of is the difference of the two individial sequences. Thus, by Theorem 1. Q.E.D.