Let such that . Prove that .

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# What goes up must come down

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4 thoughts on “What goes up must come down”

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Let such that . Prove that .

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This may be total nonsense since I’ve forgotten all my real analysis, but doesn’t this follow from the density of in the Sobolev space ?

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Actually I donâ€™t remember the definition of the Sobolev spaces, but everything is possible!

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Some details: the Sobolev space is the space of functions with weak derivative also in . In particular, any as in the problem statement is in . This space is a Banach space with norm given by .

Since is dense in , we have a sequence of smooth, compactly supported functions converging in to . Notice that this convergence implies convergence of both and . Since the integral of vanishes by Barrow’s rule, the result follows passing to the limit.

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Another proof is the following: We have so there exist two sequences of real numbers , such that for every and , .

By the Lebesgue's Dominated Convergence Theorem and Barrow's rule it follows that .

Corollary: Such function tends to at infinity. Counterintuitively, the hypothesis is necessary as you can check by constructing a counterexample.

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