Sure! Here is the answer without using LaTeX: To verify the completeness of the set of continuous functions on the interval [a, b] with the supremum norm, we need to show that every Cauchy sequence of continuous functions (with respect to the supremum norm) converges to a continuous function. Let {f_n} be a Cauchy sequence in the space of continuous functions on [a, b], with the supremum norm defined as the maximum absolute value of f(x) on [a, b]. By the definition of a Cauchy sequence, for any positive epsilon, there exists an N such that for all m, n greater than N, the supremum norm of f_n minus f_m is less than epsilon. This means that for each x in [a, b], the sequence {f_n(x)} is a Cauchy sequence of real numbers. Because the real numbers are complete, {f_n(x)} converges to some limit, which we call f(x). Define f(x) as the pointwise limit of f_n(x). Next, we need to show that f is continuous. Since the convergence is uniform (because the sequence is Cauchy in the supremum norm), the limit function f is continuous. Therefore, every Cauchy sequence of continuous functions converges to a continuous function (in the supremum norm), so the set of continuous functions on [a, b] with the supremum norm is complete. This result uses the fact that the real numbers are complete and that uniform limits of continuous functions are continuous.