Could someone please check my work and make sure I'm using the correct Lemma and theorems

Theorem 5.1.13 Let f : D -+ R and g: D -+ R and let c be an accumulation pint of D . If lim f(x) = L, lim g(x) = M , and k E R , then lim (f + 9)(x) = L + M, lim (fg)(x) = LM , and lim (kf)(x) = KL . Furthermore, if g(a) * 0 for all a E D and M * 0 , then lim a..... Theorem 4.2.1 Suppose that (S, ) and (t. ) are convergent sequences with lim S, = s and lim t = t . Then (a) lim (Sn + tn) = s + t (b) lim (ksn) = ks and lim (k + 8) = k + 8, for any k E IR (c) lim (sn . tn) = st (d) lim Sn = , provided that tn # 0 for all n and t # 0 Theorem 5.2.2 Let f : D -+ R and let c E D . Then the following three conditions are equivalent. (a) f is continuous at c (b) If (an) is any sequence in D such that (In) converges to c, then lim f(an) = f(c) (c) For every neighborhood V of f(c) there exists a neighborhood U of c such that f(UnD) CV. Furthermore, if c is an accumulation point of D , then the above are all equivalent to (d) f has a limit at c and lim f(x) = f(c) . Lemma 5.3.5 Leet f: [a, b] - R be continuous and suppose that f(a) 0 and f(1) = 21 - 3(1) = - 1