Proof for Theorem 14.8 for reference:

Use a similar approach that is used in the above proof. The only answer required is proving that e^m is irrational, using a similar approach to the above proof. 

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Prove that for all m e N, the number em is irrational. Instructions: Follow the argument in the proof of Theorem 14.8 of the notes (but add details). Use the same function 1 f(z) = "(1– a)", - T) but consider now the integral ema f(x) dx. Construct a polynomial g(x) satisfying d (emg(x)) = em" f(x). dx Assuming that em = p/q, argue that q · 2n+1 . Į is an integer, while on the other hand this expression is in (0, 1) for large n, to get a contradiction. Prove that for all m e N, the number em is irrational. Instructions: Follow the argument in the proof of Theorem 14.8 of the notes (but add details). Use the same function 1 f(z) = "(1– a)", - T) but consider now the integral ema f(x) dx. Construct a polynomial g(x) satisfying d (emg(x)) = em" f(x). dx Assuming that em = p/q, argue that q · 2n+1 . Į is an integer, while on the other hand this expression is in (0, 1) for large n, to get a contradiction.

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