I am trying to understand apostol's argument confirming the integral area of a parabolic segment.
He get's to:
There are three possibilities.
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:
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?