Exercise 7 (12 points). Let f: R" gt; Rm be continuous, where we equip R" and R" with the Euclidean metric d2. Prove that the following conditions...

Let f : R n → R m be continuous, where we equip R n and R m with the Euclidean metric d2. Prove that the following conditions are equivalent: (i) For every bounded set T ⊂ R m, we have that f −1 (T) is bounded. (ii) For every compact set T ⊂ R m, we have that f −1 (T) is compact. [Hint: To show that (ii) implies (i) you may have to prove that cl(f −1 (T)) ⊂ f −1 (cl(T)).]

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Exercise 7 (12 points). Let f: R” —> Rm be continuous, where we equip R” and R“ with theEuclidean metric d2.Prove that the following conditions are equivalent: (i) For every bounded set T C R”, we have that f_1(T) is bounded.(ii) For every compact set T C R”, we have that f ’1(T) is compact. [Hint To Show that (ii) implies (i) you may have to prove that cl( f ’1(T)) C f’1(cl(T)).]