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A "new to me" proof that $\sqrt{2}$ is irrational, found by Sergey Markelov while still in high school.

In the decimal system, a square of an integer may only end in 0, 1, 4, 5, 6, or 9, whereas twice a square may only end with 0,2,8. So if a²=2b², both a and b must end with 0.

This triggers an infinite descent which proves that this is impossible, and so a²=2b² has no solutions in integers, hence 2 is never the square of a rational.