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

He get's to:

for every

There are three possibilities.

or or

I am not really sure about this. I would get , 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 is true. From the second initial inequality we have:

for every

Divide both sides and multiply to get:

But this is obviously false when

(*)

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

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.