Intro Analysis - Real Numbers - Inequality proof

In summary, for 0<x<y, we have shown that x<\sqrt{xy}<1/2(x+y)<y by using the arithmetic/geometric mean inequality and transitive property of inequalities. This demonstrates that (\sqrt y - \sqrt x)^2>0 and provides a suggestion for organizing the proof.
  • #1
brntspawn
12
0
For 0<x<y, show that x<[tex]\sqrt{xy}[/tex]<1/2(x+y)<y

I have no difficulty showing that x<[tex]\sqrt{xy}[/tex] and 1/2(x+y)<y. I am having difficulty with [tex]\sqrt{xy}[/tex]<1/2(x+y).

x<y
xx<xy
x[tex]^{2}[/tex]<xy
x<[tex]\sqrt{xy}[/tex]

and

x<y
x+y<y+y
x+y<2y
1/2(x+y)<y
 
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  • #2
What do you know about the sign of [tex] (\sqrt y - \sqrt x)^2 [/tex]?
 
  • #3
[tex](\sqrt y - \sqrt x)^2[/tex]>0

From this I can get:
x<y
[tex]\sqrt{x}[/tex]<[tex]\sqrt{y}[/tex]
[tex]\sqrt{x}[/tex]-[tex]\sqrt{x}[/tex]<[tex]\sqrt{y}[/tex]-[tex]\sqrt{x}[/tex]
0<[tex]\sqrt{y}[/tex]-[tex]\sqrt{x}[/tex]
0<([tex]\sqrt{y}[/tex]-[tex]\sqrt{x}[/tex])^{2}
0<y-2[tex]\sqrt{xy}[/tex]+x
2[tex]\sqrt{xy}[/tex]<y+x
[tex]\sqrt{xy}[/tex]<1/2(y+x)

Assuming I am correct with this I have two more questions:
1. How did you know to work with [tex](\sqrt y - \sqrt x)^2[/tex]? Is this something I would pick up with more experience, or should I have worked backwards and manipulated it until I got back to x<y?
2. I am still relatively new to proofs, what would be the best way to put all this information together into one proof?
 
  • #4
Your question 1: I could say experience, or work backwards. I forget (it's in the distant past) where I first saw this. It is a special case of the "arithmetic/geometric mean" inequality: I don't like using wikipedia, but a brief link is

http://en.wikipedia.org/wiki/Inequality_of_arithmetic_and_geometric_means

Your problem corresponds to the case [tex] n = 2 [/tex].

Question 2: Asking about the 'best way' to organize a proof can lead to an almost infinite number of suggestions, so don't be surprised if that happens in this case. I'm in the camp that says the simplest, most direct, method, is what I want to see from students, as it shows they not only understand the concepts but are able to organize the required steps in the correct order. Here, if I can be bold enough to suggest, organize your work in three stages:
First, show the left-most inequality
Second, show the inequality you just worked through
Third, show the right-most inequality

This means you've just demonstrated that:
a < b, b < c, and c < d

since inequalities are transitive, you can put these together to show a < b < c < d.

A good part of the point of an introductory analysis course (since that's the title of your post, I assume that's your class) is training in constructing solid, concise, thorough proofs (the analysis part is important, too :) ).

Good work - and good luck.
 
  • #5
Thanks for the help. :)
 

1. What is the purpose of an inequality proof?

An inequality proof is used to show that a statement is true for all values within a certain range or set. It is often used to compare two quantities and prove that one is always greater than or less than the other.

2. What are the basic steps of an inequality proof?

The basic steps of an inequality proof include stating the given information, choosing a variable to represent the quantities involved, manipulating the inequality using properties of real numbers, and providing a logical explanation for each step.

3. How do you know which properties of real numbers to use in an inequality proof?

The properties of real numbers that are most commonly used in inequality proofs include the commutative, associative, and distributive properties, as well as the properties of inequality (such as adding or subtracting the same quantity to both sides of the inequality). The specific properties to use will depend on the given inequality and the desired outcome.

4. Can you use algebraic expressions in an inequality proof?

Yes, algebraic expressions can be used in an inequality proof. In fact, they are often used to manipulate the inequality and show the relationship between the quantities involved.

5. How can you check if your inequality proof is correct?

To check if an inequality proof is correct, you can substitute different values for the variable and see if the inequality holds true for each value. You can also double check your steps and make sure they follow the rules and properties of real numbers.

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