1. Prove that √5 is irrational

Assume that √5 is rational.

Then √5 = p/q, where p and q are co-prime integers and q ≠ 0.

Squaring both sides:
5 = p² / q²
⇒ 5q² = p²

So p² is divisible by 5, hence p is divisible by 5. Let p = 5k.

Substituting:
5q² = 25k²
⇒ q² = 5k²

Thus q is also divisible by 5, which contradicts the fact that p and q are co-prime.

Therefore, √5 is irrational.