Another $\epsilon$-$\delta$ proof of $\lim_{x \to a} x^2 = a^2$

Question

For the function $x^2$, I was curious if I can prove the limit without the assumption of $|x-a| < 1$, I reached this equation, I don't know if it's right or wrong but I tried a lot of values and I tried to visualize it and it does seem to be true, of course that's not a proof but first how can this equation be wrong?

$$|x-a| < \sqrt{a^2+\epsilon} - |a| $$

and if it's right how can I prove it?