<s>[INST] How to find the error in a proof? (that $1=0$) So I devised this proof that $1=0$. Of course it is false, but I don't know why. Why?
$$\begin{align*}
x+1&=y\\
\frac{x+1}{y}&=1\\
\frac{x+1}{y}-1&=0\\
\frac{x+1}{y}-\frac{y}{y}&=0\\
\frac{x-y+1}{y}&=0\\
x-y+1&=0\\
x-y+1&=\frac{x-y+1}{y}\\
y(x-y+1)&=x-y+1\\
y&=1\\
x+1&=1\\
x&=0\qquad * * * *\\
y-1&=x\\
\frac{y-1}{x}&=1\\
\frac{y-1}{x}-1&=0\\
\frac{y-1}{x}-\frac{x}{x}&=0\\
\frac{y-x-1}{x}&=0\\
y-x-1&=0\\
y-x-1&=\frac{y-x-1}{x}\\
x(y-x-1)&=y-x-1\\
x&=1\qquad * * * *\\
1&=0\\
\end{align*}$$
 [/INST] You have a problem already by going from 
$$x-y+1=0= y(x-y+1)$$
to
$$y=1$$
because you are dividing by zero. The fact that you are deducing $y=1$ from a starting place, namely $$x+1=y$$
that does not assume anything about $x$ or $y$, should have been a signal that something was wrong.
</s>