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Proofs : Pairwise Real Numbers, Natural and Irrational Numbers

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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

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

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Question 1.

Suppose that
for all , where are pairwise different positive real numbers.
Then we take derivatives from (1), we get
Keep doing this, we can get

Since are pairwise different positive real numbers, solving the above n equations for ,

Note: The ...

Solution provided by:
  • BSc , Wuhan Univ. China
  • MA, Shandong Univ.
Recent Feedback
  • "Your solution, looks excellent. I recognize things from previous chapters. I have seen the standard deviation formula you used to get 5.154. I do understand the Central Limit Theorem needs the sample size (n) to be greater than 30, we have 100. I do understand the sample mean(s) of the population will follow a normal distribution, and that CLT states the sample mean of population is the population (mean), we have 143.74. But when and WHY do we use the standard deviation formula where you got 5.154. WHEN & Why use standard deviation of the sample mean. I don't understand, why don't we simply use the "100" I understand that standard deviation is the square root of variance. I do understand that the variance is the square of the differences of each sample data value minus the mean. But somehow, why not use 100, why use standard deviation of sample mean? Please help explain."
  • "excellent work"
  • "Thank you so much for all of your help!!! I will be posting another assignment. Please let me know (once posted), if the credits I'm offering is enough or you ! Thanks again!"
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