Theorem: If a and b are rational numbers, b ≠ 0, and r is an irrational number, then a + br is irrational.
Proof: Assume that if a and b are rational numbers, b ≠ 0, and r is an irrational number, then a + br is rational.
By the definition of rational, we can substitute a and b with fractions where p, q, m, n are particular but arbitrary integers.
a = p/q b = m/n a + br = p/q + (m/n)r/1 = p/q + mr/n = (pn + qmr) / qn
Since r is irrational, we know that both the numerator and the denominator cannot be rational numbers, which implies a + br is irrational, which contradicts the fact that a + br is rational. This contradiction shows the supposition is false, therefore the theorem is true.