Proof that a quantity is greater than 1/2

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I was able to use the definition of convergence to prove that ##1 - \frac{N}{n} > \frac{1}{2}## for all ##n>N##. In summary, the conversation discusses how to prove that ##1 - \frac{N}{n} > \frac{1}{2}## given that ##n>N##. The speaker suggests using the definition of convergence and choosing a suitable value for ##\epsilon## to show the inequality. They also mention that they have solved the problem on their own.
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Mr Davis 97
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I'm looking at the quantity ##\displaystyle 1 - \frac{N}{n}##, and trying to prove that it is greater than ##1/2##, given that ##n> N##. I thought that since ##\lim_{n \to \infty} 1 - \frac{N}{n} = 1##, we could use the definition of convergence to get this inequality, for suitable ##\epsilon##. For example, from the definition of convergence we can see that ##|(1 - N/n) - 1| < \epsilon## which implies ##N/n < \epsilon##, and so ##1-N/n > 1- \epsilon##. So if we let ##\epsilon = 1/2##, we get our result. Is that how I would do it?
 
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Suppose N = 7 and n = 8. Then n > N. Then 1 - N/n = 1 - 7/8 = 1/8 < 1/2. So I don't think you will have any luck proving the statement as you've stated it.
 
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Is this related to your other thread with ##a_n = 2^{(-1)^n}## and still open? What do you want to show?
 
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fresh_42 said:
Is this related to your other thread with ##a_n = 2^{(-1)^n}## and still open? What do you want to show?
This is not related, but I actually figured out what I wanted to show for this question.
 

1. What is "Proof that a quantity is greater than 1/2"?

"Proof that a quantity is greater than 1/2" is a mathematical concept that involves showing evidence or demonstrating that a certain quantity is larger than one-half, or 0.5.

2. Why is it important to prove that a quantity is greater than 1/2?

Proving that a quantity is greater than 1/2 is important in many mathematical and scientific applications. It can help in making accurate predictions, drawing conclusions, and solving problems in various fields such as statistics, physics, and economics.

3. How do scientists prove that a quantity is greater than 1/2?

Scientists use various mathematical methods and techniques to prove that a quantity is greater than 1/2. These methods may include using logical reasoning, algebraic manipulation, and calculus to show that the value of the quantity is indeed larger than 0.5.

4. Can a quantity be greater than 1/2 without being explicitly proven?

Yes, it is possible for a quantity to be greater than 1/2 without being explicitly proven. In some cases, the value of the quantity may be known through prior experiments or observations, and there may be no need for further proof.

5. Are there any real-life examples of quantities being greater than 1/2?

Yes, there are many real-life examples of quantities being greater than 1/2. For instance, the probability of getting heads when tossing a fair coin is greater than 1/2, as there are only two possible outcomes (heads or tails). Another example is the percentage of students who passed an exam, which is also typically greater than 1/2.

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