My attempt: Let $x $ be a number such that $$(y+e)\le (a)^{\frac{1}{2}}$$ where $a$ is the number whose square root we are trying to find and $e>0$ so that $y+e$ is close to square root of $a$. Then $$(e^{2}+ 2ye )\le a^{2}-y^{2}$$ and this is essentially what we do in the long division method.
Am I on the right track? And what more do I need to add to make this proof complete?