- #1
ehrenfest
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Homework Statement
http://www.math.cornell.edu/~putnam/ineqs.pdf
Can someone give me a hint on problem 1?
Prove the second inequality assuming the first
- Thread starter ehrenfest
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In summary, the conversation is about a math problem, specifically problem 1 from a PDF file. The person is looking for a hint on how to solve the problem and has already been given one hint. There is also a discussion about the first and second inequality, and one person realizes that the expression given in the problem is already in a form where the first inequality can be applied. The other person admits to feeling stupid for not realizing this sooner.
http://www.math.cornell.edu/~putnam/ineqs.pdf
Can someone give me a hint on problem 1?
- #2
Hurkyl
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You were already given a hint: prove the second inequality assuming the first. Second hint below:
Isn't the expression
[tex]\left( \frac{ \frac{1}{a_1} + \cdots + \frac{1}{a_n} }{n} \right)^{-1}[/tex]
already in a form to which the first inequality is applicable?
Isn't the expression
[tex]\left( \frac{ \frac{1}{a_1} + \cdots + \frac{1}{a_n} }{n} \right)^{-1}[/tex]
already in a form to which the first inequality is applicable?
Last edited:
- #3
ehrenfest
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Yes. I read the problem. I spent 30 minutes manipulating both the first and the second inequality. Maybe I am missing something obvious, but I just don't see how to manipulate correctly. I tried logarithms, finding common denominators...
- #4
Hurkyl
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Bah, you saw it before I edited my post. 
- #5
ehrenfest
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Wow. I feel stupid now. :)
FAQ: Prove the second inequality assuming the first
1. What does it mean to "prove the second inequality assuming the first"?
Proving the second inequality assuming the first means to show that if the first inequality is true, then the second inequality must also be true. This is a common technique used in mathematical proofs to build upon previous assumptions or results.
2. Why is it important to prove both inequalities?
Proving both inequalities helps to establish a stronger argument and provides a more complete understanding of the relationship between the two quantities involved. It also ensures that both statements are valid and supported by evidence.
3. How do you go about proving the second inequality assuming the first?
To prove the second inequality assuming the first, you must use logical reasoning and mathematical techniques such as algebra, geometry, or calculus. You will need to carefully examine the given information and use the properties of the first inequality to make a logical argument for the second inequality.
4. Can you provide an example of proving the second inequality assuming the first?
Sure, let's say we are given the first inequality "x + 2 < 10". To prove the second inequality "x < 8", we can subtract 2 from both sides of the first inequality to get x < 8, which satisfies the second inequality. Since the first inequality is true, we can assume that the resulting inequality is also true.
5. How does proving the second inequality assuming the first relate to the overall proof?
Proving the second inequality assuming the first is just one step in the overall proof. It helps to establish the validity of the second inequality and strengthens the overall argument. By connecting the two inequalities, it shows a clear and logical progression in the proof.