Question

For any positive real number x,show that there exists an irrational number r such that 0

Answer 1

Step-by-step explanation:

If x is irrational, then y = x/2 is also an irrational number such that 0 < y < x If x is rational,

then y = x/√2 is an irrational number such that

$y = \frac{x}{ \sqrt{2} } < x \: (as \: \sqrt{2 \: } < 1)$

Hence, for any positive real number x, there exists an irrational number y such that 0 < y < x.