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!