Prove that, if u is harmonic and bounded on C, then u is constant. (Hint: Use Theorem 6.6 and Liouville's Theorem (Corollary 5.13).) Theorem 6.6 states that: Suppose u is harmonic on a simply-connected region G. Then there exists a harmonic function v in G such that f = u + iv is holomorphic in G. Liouville's Theorem states that: Any bounded entire function is constant.

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