Could someone please check my work and let me know if I am correctly using all of the included theorems that are referenced in the proof

Please state all definitions and theorems that you will need: 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(x) * 0 for all at E D and M # 0 , then lim ( ; ) (2 ) = M. Theorem 4.2.1 Suppose that (S, ) and (tn) are convergent sequences with lim S, = s and lim tn = t . Then (a) lim (sn + tn) = s + t (b) lim (ksn) = ks and lim (k + s,) = k + s , for any k E R (c) lim (Sn . tn) = st (d) lim (am ) = , provided that in # 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 (2n) 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) . Theorem 5.3.6 (Intermediate Value Theorem) Suppose that f : [a, b] - IR is continuous. Then f has the intermediate value property on [a, b] . That is, if k is any value between f(a) and f(b) [i.e., f(a)