$x+\dfrac{1}{y}=y+\dfrac{1}{z}=z+\dfrac{1}{x}$, prove that $xyz=1$ Let $x>0$, $y>0$ and $z>0$ such that $x\neq y $ , $y\neq z$ and $x\neq z$. If $x+\dfrac{1}{y}=y+\dfrac{1}{z}=z+\dfrac{1}{x}$, prove that $xyz=1$

Hint: Notice that
\begin{align} x-y &= \frac{y-z}{yz} \\
y-z &= \frac{z-x}{xz} \\
z-x &= \frac{x-y}{xy}.
\end{align}
Now multiply.
