Prooving that $sign(x)=1$(?)

Question

Okay so I was having some thoughts about the $sign$ of real numbers and here is what I have:

$$(1) \forall x \in \Bbb R$$ $$(2) sign(x)= \frac{x}{|x|}$$ $$(3) |x|= \sqrt{x^2}$$ $$(4) sign(x)= \frac{x}{\sqrt{x^2}}$$ $$(5) sign(x)*\sqrt{x^2}=x$$ $$(6) \sqrt{x^2*sign(x)^2}=x$$ $$(7) x^2*sign(x)^2=x^2$$ $$(8) sign(x)^2= \frac{x^2}{x^2}$$ $$(9) sign(x)^2 =1$$ $$(10) sign(x)=1$$

The entire thing can be summarized as: $$sign(x)^2=1$$ $$sign(x)=1$$

Here are my thoughts: this is probably a classic case of square/square root information destruction. In an example where information is not destroyed in the transition from line 9 to 10 there would be a $\pm$ leading to $sign(x)=\pm 1$ which is true for all $x \not = 0$.

Another thing is that line 6 is problematic because it implies that $|x|=x$ for all nonzero $x$, further implying that $x>0$ which is not nessacrily true.

Answer 1

Error $1$ in step $(2)$ occur if $x=0$ . In that case you have to define $sign(x)=\frac{x}{\mid x\mid}$ if $x\ne0$ and $sign(0)=0$

Error $2$ is in step $(6)$ as if $x<0$ this doesn't happen.

Error $3$ is in step$10$ as $(sign(x))^2=1\Rightarrow sign(x)=1,-1$

Answer 2

Error (5) to (6) :How can you be sure that $sign(x) >0 \to sign(x)=\sqrt{(sign(x))^2}$
for example try (6) for $x=-2 $ you will see (6) is not true.