Proof of (ir)rational numbers between real numbers a and b

[PcumP_Ravenclaw]

Q4) Let a and b be real numbers with a < b. 1) Show that there are infinitely
many rational numbers x with a < x < b, and 2) infinitely many irrational
numbers y with a < y < b. Deduce that there is no smallest positive
irrational number, and no smallest positive rational number.

1)
a < x < b
a + (b - a)/n or b - (b - a) /n

n = Natural numbers, N
n = 0 to infinity

This shows that there are infinitely many rational numbers between a and b which can be written in the form of integers

Deduce that there is no smallest positive rational number.

This is because say a = 0 and b = 1. Note 0 is not a positive or negative number it has no sign.

so 0 < x < 1

as there are infinitely many n. there is no smaller number e.g.

0 + (1 - 0) / n

it starts from 1 when n = 1 and decreases to very very small... but never stops... as positive numbers keep decreasing

Please check if my proof above is correct and suggest how I can prove that there are infinitely many irrational numbers between two real numbers?

Danke!

 

[mathman]

Your first proof is incorrect. What you proved is that there are an infinite number of real numbers between a and b. You don't know whether they are rational or irrational. It will take a little work to do what you need. An approach would be to find a rational number in the interval and use rational increments, remaining in the interval. For an infinite number of irrationals, start with an irrational but still use rational increments.