I know this proof is correct but could someone please answer my questions in black ink

for all x E R . A function f: R - R is said to be periodic if there exists a number k > 0 such that f(a + k) = f(x) Suppose that f : R - R is continuous and periodic. Prove that f is bounded and uniformly continuous on R . Theorem: Every continuous periodic function is bounded and uniformly continuous on R. Proof: Suppose that f : R - R is a continuous and periodic function with period k > 0. Let g be the restriction of f on [0, k], i.e, g(x) = f(x) for all a E [0, k). Then g is continuous on the compact set [0, k.]. Therefore, by theorem 5.4.6, it is bounded and uniformly continuous on [0, k.]. Thus, for given & > 0, there exists o > 0 such that whenever |x - y)