Not able to solve the below mentioned inequality. Someone please explain me it's solution.

Question

This is an in equality with a solution given below. I'm not able to understand it. It will be very helpful if someone can help me understand it. Thanks.

The inequality is in the image attached with this question.

Answer 1

1. Right hand inequality. Let $f=\frac{1}{2}-\frac{3}{n}$.

Note that if $n\geq 1$ then $f\leq \frac{1}{2}$. This is because for $n=1$, $f=\frac{-5}{2}$. Now, when you increase $n$ the term $\frac{3}{n}$ gets "smaller" as $n$ gets "bigger". So for very big $n$, the term $f$ is closer to $\frac{1}{2}$. So you can say that for $n\geq 1$ the maximum value of $f$ is $\frac{1}{2}$ and the minumum value is $\frac{-5}{2}$.

1. The left part is quite similar, give it a try.

Answer 2

For the right hand side of the inequality (with c$_2$), you see that for any natural number n (i.e. n=1,2,3,4....), the quantity 3/n is always positive, so 1/2 minus a positive quantity will always be less than 1/2, right?

For the left hand side, you see that when n $\geq$ 7, you get 1/2 - 3/7 = 1/14. Again as n becomes greater than 7, this number will get smaller right? (Try this: n > 7 implies 1/n < 1/7 implies 3/n < 3/7 implies -3/n > -3/7 implies 1/2 - 3/n > 1/14)