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Proofs : Pairwise Real Numbers, Natural and Irrational Numbers

Problem 1. Let n be a natural number and a1.... ,an > 0 be pairwise different positive real numbers. Show that if λ1...λn are such real numbers that the equality
...
holds true for all x E R then ....

Problem 2. Show that there are infinitely many real numbers x in the interval [0, pi/2] such that both sinx and cosx are rational numbers.
Problem 3. Show that. for any real number x E [?1, 1] and any positiVe real number ε > 0 there exists a natural number n such that
....

Please see the attached file for the fully formatted problems.

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Solution Summary

Proofs involving Pairwise Real Numbers, Natural and Irrational Numbers are provided. The solution is detailed and well presented.

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