# Proofs : Pairwise Real Numbers, Natural and Irrational Numbers

Not what you're looking for?

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.

##### Purchase this Solution

##### Solution Summary

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

##### Solution Preview

Please see the attached file for the complete solution.

Thanks for using BrainMass.

Question 1.

Suppose that

.............................(1)

for all , where are pairwise different positive real numbers.

Then we take derivatives from (1), we get

..................(2)

Keep doing this, we can get

...................(3)

.....................

...............(n)

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

Note: The ...

###### Education

- 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!"
- "Thank you"
- "Thank you very much for your valuable time and assistance!"

##### Purchase this Solution

##### Free BrainMass Quizzes

##### Exponential Expressions

In this quiz, you will have a chance to practice basic terminology of exponential expressions and how to evaluate them.

##### Geometry - Real Life Application Problems

Understanding of how geometry applies to in real-world contexts

##### Multiplying Complex Numbers

This is a short quiz to check your understanding of multiplication of complex numbers in rectangular form.

##### Graphs and Functions

This quiz helps you easily identify a function and test your understanding of ranges, domains , function inverses and transformations.

##### Know Your Linear Equations

Each question is a choice-summary multiple choice question that will present you with a linear equation and then make 4 statements about that equation. You must determine which of the 4 statements are true (if any) in regards to the equation.