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You are given three real numbers a, c, d, with c < a < d, and a set of real numbers, D, such that the open interval (c, d) is a subset of D. 1. Suppose {an} is a sequence in D converging to a. Prove that there exists an index N such that for all n ≥ N, {an} is in (c, d). 2. Prove that there is a subsequence of {an} in (c, d) which converges to a. 3. You are given a real-valued function f on D. Let g denote the restriction of f to the open interval (c, d). Using the sequential definition of continuity, show that f is continuous at a iff g is continuous at a.