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If $$x+\frac{1}{y}=y+\frac{1}{z}=z+\frac{1}{x}$$ then prove $$x= y= z$$

I cannot solve this problem. Please help me out. This is from ratio and proportion example.

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$x+\frac{1}{y}=y+\frac{1}{z}$

$xyz+z=y^2z+y$ (1)

$y+\frac{1}{z}+z+\frac{1}{x}$

$xuy+x=z^2x+z$ (2)

$x+\frac{1}{y}=z+\frac{1}{x}$

$xyz+y=x^2y+x$ (3)

add equations (1),(2)&(3)

$3xyz=x^2y+z^2x+y^2z$

so, $x=y=z$

Let $x=1, \;y=-\frac{1}{2},\; z=-2$.

Then $x+\frac{1}{y}=y+\frac{1}{z}=z+\frac{1}{x}=-1$, so the assertion as stated is not correct.

Assume WLOG that $x$ is the smallest of your three numbers.

Now suppose that $x<z$. Then

$$x+\frac{1}{y}<z+\frac{1}{y}\leq z+\frac{1}{x},$$which is a contradiction; hence $x=z$, and together with your second equality, this implies that $z=y$.

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