and, consequently, infinitely many. I am new to proofs so could you please check if this proof is correct? Let x be an irrational number in the interval In = [an, bn], where an and bn are both rational numbers, in the form p/q. Let z be the distance between x and an, So: x - an = x - p1/q1 = x + (-p1/q1) But an irrational number (in this case x) + a rational number is also an irrational number. Therefore distance from an to x, which is z, is irrational. The distance from x to the origin, is an + z. But again, an irrational number plus a rational number is also irrational. Therefore, there is always at least one rational number between any two rational numbers. However, the same proof can be applied to an infinite amount of subintervals within In, therefore there is an infinite amount of irrational numbers as well.