The mathematician Leibniz based his calculus on the assumpti

The mathematician Leibniz based his calculus on the assumption that there were infinitesimals, positive real numbers that are extremely small: smaller than all positive rational numbers certainly. Some calculus students also believe, apparently, in the existence of such numbers since they can imagine a number that is just next to zero. Is there a positive real number smaller than all positive rational numbers? Show how you might prove your assertion using terms a calculus student would understand.

Solution

Solution :- We have a question that \" Is there a positive real number smaller than all positive rational numbers\".

Suppose there is a smallest positive real number x.

x is such that , x > 0 and x |R.

Let y = x/10

Then we get contradiction.

This implies that y < x.

==> We can always construct a rational number that is less that the smallest positive real number.

This is process has no end.

So there is no positive real number smaller than all positive rational numbers.