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mathoverflow.net/questions/5312
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That’s why, when I prouve Liouville theorem in my class, I explain that this looks like a useless theorem, though we shall see its power by proving the Fundamental Theorem of Algebra by using it.
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A good reminder to myself that when giving an "example" of a theorem, it's better to give an example of the usefulness of the theorem than of an object satisfying the hypotheses. For example, "Even though sin(x) is a bounded real-analytic function, the complex version cannot be."
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