If look

here.
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.