Could someone please check my work and let me know if my circled wording is OK

Please state all definitions and theorems that you will need: Definition 5.4.1 Let f : D - R . We say that f is uniformly continuous on D if for every & > 0 there exists a 6 > 0 such that If(x) - f(y)| 0 such that f(a + k) = f(a) for all a E R . Suppose that f : R - R is continuous and periodic. Prove that f is bounded and uniformly continuous on R . WTS that a continuous periodic function, f , on R is bounded and uniformly continuous on R . Suppose that f : R - R is a continuous and periodic function with period k > 0 . Let g be the restriction of f on the interval, [0, k] denoted by f [o,k] , such thatg: [0, k] - RR, g(a) = f(x). Since |k - 0| 0, then [0, k] is bounded both above and below by the definition of boundedness. Since [0, k] contains all its boundary points, the interval is closed by the definition of a closed set, and since it is closed and bounded, [0, k] is compact by the Heine Borel Theorem 3.5.5. Since f is continuous on R , g is continuous on the compact subset, [0, k], and thus, by Theorem 5.4.6, g is uniformly continuous on this restricted interval of f . Also, since the restricted interval is bounded, g is bounded. Since f is periodic with period k , Vx E R , sup f(x) | = suplg(x) | , so f is also bounded by the definition of maximum and minimum. Since g is uniformly continuous on the restricted interval, then VE > 0, 3 > 0 such that whenever x - yl