I need help figuring out my mistake. Proof. Proving by contradiction that $\sqrt{\frac{1}{2}}$ is irrational. Suppose $\sqrt{\frac{1}{2}}$ is rational so: $\sqrt{\frac{1}{2}}=\frac{m}{n}$. Where $m/n$ are in lowest terms.

Squaring both sides and solving for $n$ we have $n^{2}=2m^{2}$.

So $n^2$ is even therefore $n$ is also even therefore $n=2k$ and $n^{2}=4k^{2}$.

Now, replacing $n^2$ in $n^{2}=2m^{2}$, we have $4k^{2}=2m^{2}$. Which means $m$ is also even.

So we are left with an even numerator and an even denominator which contradicts the initial assumption that m/n are in lowest terms. Therefore $\sqrt{\frac{1}{2}}$ is irrational. (which we know is wrong)