I need to find the flaw in the following proof:

$a,b\in\mathbb{R}$\ $\left\{ 0 \right\} $ such that $a=b$

1) Multiplying both sides by $a$ yields the equality: $a^2=ab$

2) Subtracting $b^2$ from both sides yields the equality $a^2-b^2=ab-b^2$

3) Then, $a^2-b^2=ab-b^2\Rightarrow (a-b)(a+b)=b(a-b)$

4) Then, dividing both sides of $(a-b)(a+b)=b(a-b)$ by $(a-b)$ yields: $a+b=b$

5) Substituting $a=b$ yields $2b=b$

5) Therefore, $2=1$

**My thoughts and ideas on this:**

It seems to me that step $5$ has an error. It should be that $a+b=b\Rightarrow a=0$. However, we initially said that $a,b$ cannot be $0$.