I am trying to understand apostol's argument confirming the integral area of a parabolic segment.

He get's to:

$\displaystyle \frac{b^3}{3}-\frac{b^3}{n} < A < \frac{b^3}{3}+\frac{b^3}{n}$ for every $\displaystyle n \ge 1$

There are three possibilities.

$\displaystyle A > \frac{b^3}{n}$ or $\displaystyle A = \frac{b^3}{n}$ or $\displaystyle A < \frac{b^3}{n}$

I am not really sure about this. I would get $\displaystyle -\frac{b^3}{n} < A < \frac{b^3}{n}$, so why does he get three possibilities. I understand any number is either less, equal or greater, but its still a little confusing.

The next contradiction I think I sort of understand.

Suppose $\displaystyle A > \frac{b^3}{n}$ is true. From the second initial inequality we have:

$\displaystyle A-\frac{b^3}{3} < \frac{b^3}{n}$ for every $\displaystyle n \ge 1$

Divide both sides and multiply to get:

$\displaystyle n < \frac{b^3}{A-\frac{b^3}{3}}$

But this is obviously false when

(*) $\displaystyle n \ge \frac{b^3}{A-\frac{b^3}{3}}$

Ok, clearly that is true, but why is it a contradiction that

$\displaystyle A > \frac{b^3}{n}$

Is it because (*) is always a fraction and can be at most 1 therefore n cannot be any less than that becuase n is bigger than one?

Thanks

Regards

Craig.