Prove there exists a real number $x$ such that $xy=y$ for all real $y$ 
Prove: There exists a real number $x$ such that for every real number $y$, we have $xy=y.$

In class I learned that I can prove a statement by:


*

*proving the contrapositive,

*proof by contradiction,

*or proof by cases.


Can I do something along the lines of $xy=y$
divide both sides by $x$ to get $y=\frac{y}{x}$
which would then be equal to $y=\frac{y}{x}=xy$
divide everything by $y$ to get $1=\frac{1}{x}=x$
so $x=1$?

There exists a real number $x$ such that for every real number $y$, we have $xy=x$

I know I somehow have to prove $x=0$ right? But I am not sure how to go about this.

Since $\mathbb R \setminus \{0\}$ is a group (wrt. multiplication), the identity satisfies this equation. The identity is clearly $1$.
