Could someone please check my work and let me know if it's correct I'm very unsure of my work

Please state all definitions and theorems that you will need: Thearea 5.4/ Suppose of: D - R is continuous on a compact set is. Then of is uniformly continuous on D. for all a E R. A function f: R -+ R is said to be periodic if there exists a number k > 0 such that f(x + k) = f(x) Suppose that f : IR - R is continuous and periodic. Prove that f is bounded and uniformly continuous on R . Suppose That fill- R is continuous and periodic. Consider The periodic function q: 50, KJ- PRO, 9 (x) = f(x) = f(x+K). Since 10, K] is a closed and bounded interval, 9 is bounded on Co, K] and Thus uniformly continuous or This restricted interval by Theorem 5.4.6. To show that of is uniformly continuous on FR. Let ESO. Then 780 9 (gla)-g ( b)/