(b) Let f: (X1, d1) ? (X2, d2) be a continuous function and let K ? X1 be a compact set. Prove that f(K) is compact. (c) Give an example of a function f: (X1, d1) ? (X2, d2) that sends compact sets to compact sets, but is not continuous. (d) Let ?³ be equipped with the Euclidean metric. Prove that {(x, y, z) | x² + y² + z² = 1} is a compact subset of ?³. (e) Define the concept of a Cauchy sequence (xn) in a metric space X. Let Y ? X. Define what it means for Y to be a complete subset. (f) Prove that every compact subset Y of a metric space X is complete.
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Let (y_n) be a sequence in f(K). Since f is continuous, we know that for each n, there exists x_n in K such that f(x_n) = y_n. Since K is compact, there exists a subsequence (x_n_k) of (x_n) that converges to some x in K. Since f is continuous, we have f(x_n_k) Show more…
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