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**novice** Prove that $\displaystyle \forall x \in \mathbb{R}[\exists y \in \mathbb{R}(x+y=xy)\Leftrightarrow x \not = 1]$

Let $\displaystyle P:\forall x \in \mathbb{R}[\exists y \in \mathbb{R}(x+y=xy)$.

Let $\displaystyle Q:x \not = 1$

For $\displaystyle P\Rightarrow Q$, I did this:

For $\displaystyle \forall x \in \mathbb{R} \exists y \in \mathbb{R},$,

$\displaystyle x+y=xy \Rightarrow xy-y=x \Rightarrow y=\frac{x}{x-1}\Rightarrow x \not = 1$

Now $\displaystyle Q \Rightarrow P$:

$\displaystyle x\not = 1 $ means $\displaystyle x >1$ or $\displaystyle x<1$, but the possibility is endless.

I am cheating here:

$\displaystyle x \not = 1 \Rightarrow x>1$ or $\displaystyle x<1$

Without lost of generality, say $\displaystyle x>1$. Then for some $\displaystyle y$, $\displaystyle xy >y \Rightarrow xy=y+x$. I know this is not true for all $\displaystyle x.$

Does any one have a convincing proof for this part?