Originally Posted by

**MichaelLitzky** in Apostols's Calculus, section 3.5, Proofs of Basic Limit Theorems, page 136, Proofs of (i) and (ii), he makes an assumption which I'm hoping someone can explain to me how he justifies.

He's proving the first limit theorem, that the limit of a sum is the sum of the limits. He says "Since the two statements lim f(x) = A and lim[f(x)-A]=0 are equivalent..."

I can readily see that lim f(x) - A = 0 but don't you need the second limit theorem, that the limit of a difference is the difference of the limits, to therefore conclude that lim[f(x)-A]=0?