Prove there is no natural number whose square is 2 Producing

Prove: there is no natural number whose square is 2. (Producing a decimal approximation
of sqrt(2) is not proof. You may use without proof that squaring an inequality between natural
numbers preserves the inequality.)

Solution

Assume sqrt(2) = m/n, where n and m are relatively prime, reduced to simplest form. Clearly then, m cannot be even.

Square both sides:
2 = m² / n²
2 n² = m²

m² is even, so m must also be even. Therefore assumption is false, and sqrt(2) cannot be expressed as a ratio of relatively prime integers so sqrt(2) is irrational, whose square is 2.

So, we can say that there is no natural number whose square is 2.