Could someone please check my work and make sure I correctly used the included theorems

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() * 0 for all a E D and M * 0 , then lim ( 4 ) (2 ) = M Theorem 4.2.1 Suppose that (8n) and (tn) are convergent sequences with lim s,, = s and lim t = t . Then (a) lim (Sn + tn) = s + t (b) lim (ks,) = ks and lim (k + 8,) = k + s , for any k E IR (c) lim (sn ' tn) = st (d) lim " )= 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 (In) is any sequence in D such that (In) converges to c , then lim f(In) = 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 & is any value between f(a) and f(b) [i.e., f(a) a , then if f(a) = b > a , thenf(a) > a If a f(a) - a > 0 and f(b) - b is continuous. Suppose In + C . Then by Theorem 5.1.13, lim F(In) = lim [f(xn) - Xn] = [ lim f(n)] - [ lim (an)] by Theorem 4.2.1 = f(c) - c = F(c) Thus, by Theorem 5.2.2, F is continuous at c . Thus F(a) = f(a) - a > o and F(b) = f(b) -b <. by the intermediate value theorem since f and are continuous e b such that a which means there exists between thatf c="0" is fixed point. therefore if : then has point>