Consider a sequence {xj} that converges to x, so that {f(xj)} converges to a and {g(xj)} converges to b.

Given a criterion q, q > 0, choose n such that all terms xj for j > n have

Then we get f(xj) + g(xj) -a -b< q, and we can conclude that the limit of f + g is a + b.