Perhaps you might like to try generalising this proof to prove that $\sqrt n$ is irrational when $n$ is not a square number.
Now $(1.4)^2=1.96$, so the number $\sqrt 2$ is roughly $1.4$. To get a better approximation divide $2$ by $1.4$ giving about $1.428$, and take the average of $1.4$ and $1.428$ to get $1.414$. Repeating this process, $2\div 1.414 \approx 1.41443$ so $2\approx 1.414 \times 1.41443$, and the average of these gives the next approximation $1.414215$. We can continue this process indefinitely,
getting better approximations, but never finding the square root exactly.
If $\sqrt 2$ were a rational number, that is if it could be written as a fraction $p/q$ where $p$ and $q$ are integers, then we could find the exact value. The proof sorter shows that this number is IRRATIONAL so we cannot find an exact value.