Proof that a quantity is greater than 1/2

[Mr Davis 97]

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?

 

[phyzguy]

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.

 

[fresh_42]

Is this related to your other thread with $a_n = 2^{(-1)^n}$ and still open? What do you want to show?

 

[Mr Davis 97]

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.