Square Root Is Irrational Proof

https://www.youtube.com/watch?v=meudJkjKEe8&t=0s

Proof

Assume that $\sqrt{2}\in Q$. This means that $\sqrt{2}=\frac{a}{b}$ where $a,b\in Z$ and the GCF is 1. Squaring both sides $2=\frac{a^2}{b^2}$ Rearranging we get: $2b^2=a^2$ this implies that $a^2$ is an Even Integer. This further implies that $a$ is an Even Integer (Square Has Same Parity Proof) $a=2k$ where $k\in Z$ Then rewrite: $2b^2=(2k{)}^2$ $b^2=2k^2$ This implies that $b^2$ is an Even Integer and $b$ is an Even Integer by Square Has Same Parity Proof Since $b$ and $a$ are even, they have a shared GCF of atleast 2. This contradicts our initial assumption of a GCF of 1. Therefore $\sqrt{2}\notin Q$ $\square$