Suppose that the power series 2 our" and Z bum" converge in (R, R). Let E = {:c G (-R, R) : Zens-r" = anxn}. \"E has a limit point in S, then on = b", for n = 0,1,2, ..., that is, w ix} E aux" = E bust" n=0 n=0 Proof. Set on = on b", and f(:r) = Zone", for all a: E (R, R). Then. it is clear that f(3:) = 0 on E. Let A be the set of all limit points of Bin 5', and let B = {:r: E S : .1: 51A} It is clear that B is Open in S. PROVE: A is also open. Since A and B are open in S, Am B = qb and S = A UB, then either A or B is empty. Since A is not empty (since E has a limit point), then B must be empty, and hence, S: A U B: A. Since f is continuous in S and A C E, then B = S. and thus, on = 0 for n = l, 2,3, which our desired conclusion