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517 HW1 1. Prove that for each positive real number x, there is a real number y such that y 2 = x. (Hint: let y = sup{z ∈ R : z 2 < x}, and prove that y 2 = x.) 2. Prove that a set S is infinite if and only if there is a proper subset A of S such that |A| = |S|. 3. Give an example of a bounded countable set of real numbers with a countable set of limit points. 4. Consider Q as a metric space with d(x, y) = |x − y|, and let S = {x ∈ Q+ : 2 < x2 < 3}. Prove that S is both open and closed in this metric space. 5. Prove that a sequence converges if and only if all of its subsequences converge. Also, show that if every subsequence converges, then they all converge to the same limit. 1