Order preserving bijection from $\mathbb{Q}$ to $\mathbb{Q}\backslash\lbrace{0}\rbrace$ 
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*How can one prove the existence of an order preserving bijection from $\mathbb{Q}$ to $\mathbb{Q}\backslash\lbrace{0}\rbrace$?

*Can you give an example of such a bijection? 

Choose an irrational number $\alpha$.
Let $x_1, x_2, \ldots$ be a strictly increasing sequence of rational numbers that converge towards $\alpha$.
Let $y_1, y_2, \ldots$ be a strictly decreasing sequence of rational numbers that converge towards $\alpha$.
Then define $f:\mathbb Q\to\mathbb Q\setminus\{0\}$ as:


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*$f$ maps $(-\infty,x_1]$ to $(-\infty,-1]$ by subtracting $x_1+1$ from everything.

*For every $n$, $f$ maps $[x_n,x_{n+1}]$ to $[-\frac1n,-\frac1{n+1}]$, by linear interpolation between the endpoints.

*For every $n$, $f$ maps $[y_{n+1},y_n]$ to $[\frac1{n+1},\frac1n]$, by linear interpolation between the endpoints.

*$f$ maps $[y_1,\infty)$ to $[1,\infty)$ by subtracting $y_1-1$ from everything.

