1. ## Proving inequality

Prove that for all sufficiently large n,

$\displaystyle \frac{(1+\log2)n}{\log\frac{n}{2}} \leq \frac{2n}{\log n}$

2. ## Re: Proving inequality

Well for starters, find the smallest value where the inequality holds true, then show the right hand function grows faster than the left hand function.

3. ## Re: Proving inequality

First , find by guessing some small value for which inequality holds , then prove inequality using math induction .

4. ## Re: Proving inequality

Hey Kanwar245.

Hint: Try taking log(n/2) to the RHS and use the fact that log(n/2) = log(n) - log(2).

5. ## Re: Proving inequality

Originally Posted by princeps
First , find by guessing some small value for which inequality holds , then prove inequality using math induction .
Is mathematical induction wise here? We aren't told that n only takes on integer values...

6. ## Re: Proving inequality

Originally Posted by Prove It
Is mathematical induction wise here? We aren't told that n only takes on integer values...
Common notation for integer variable is a letter "n" .

7. ## Re: Proving inequality

Originally Posted by Prove It
Well for starters, find the smallest value where the inequality holds true, then show the right hand function grows faster than the left hand function.
How can I show that the function on the right side grows faster than the function on the left side?

8. ## Re: Proving inequality

Originally Posted by Kanwar245
How can I show that the function on the right side grows faster than the function on the left side?
Take the derivative of each side. Show the right hand derivative is greater than the left.