I have a method no epsilon-delta already solved but looking for additional support using lemma 3.2.8 and epsilon-delta method

Theorem 3.2.10 Let I C R be an open interval, let c E I, let f, g: 1-{C} - R be functions and let k E R. Suppose that lim x->c f(x) and lim x->c g(x) exist 1. lim x->c [f+g] (x) exists and lim x->c[f+g](x) = lim x->cf(x) + lim x->cg(x) For this Theorem we will be utilizing Lemma 3.2.8 Let I CR be an open interval, let c E I and let f.g: I-{c)>R be functions. Suppose that lim f(x) =0, and that g is bounded. Then lim f(x)g(x) = 0. Proof by epsilon-delta method