newtonian mechanics - Is this derivation for schwarzschild radius for a black hole of mass $M$ correct?

Consider a body of mass $M$. We know that light can’t escape a black hole. Speed of light being the highest possible could be set as the escape velocity.(??) Then

$$\text{Escape velocity}^2=(2GM/r)$$ Solving for $r$ we get $$r=2GM/v^2$$ Since $v=c$; $$r=2GM/c^2$$ My only problem with this derivation is that shouldn’t we be using relativistic mechanics instead of newtonian? If we do use Relativistic mechanics,is there any proof similar to this one?