Monotone increasing bijection from $\mathbb{R}$ to $(0,1)$. Give an example of a monotone increasing function $f: \mathbb{R} \to (0,1)$ such that $f$ is a bijection.
I have an example in mind that follows $$g(x)=\frac{1}{1+e^x} ,x \in \mathbb{R}.$$ Then $g$ is a monotone decreasing bijection from $\mathbb{R}$ to $(0,1)$.  
Then consider the function $f: \mathbb{R} \to (0,1)$ as $$f(x)=1-g(x).$$ Then is it the required example?
Is my answer is correct that, above define $f$ is a monotone increasing bijection from $\mathbb{R}$ to $(0,1)$?
If this example is wrong, please suggest me an appropriate example.
Thanks.

Your example is correct.
You may take other bijective function as $$f:(-\infty,\infty) \rightarrow (0,1),
f(x)=\frac{1}{2}(1+\mbox{erf}(x)), \mbox{erf}(x)=\frac{2}{\sqrt{\pi}}\int_{0}^{x} e^{-t^2} dt,$$ then $$f'(x)=\frac{2}{\sqrt{\pi}} e^{-x^2}>0, f(\pm \infty)=\pm 1.$$
Another example is $$f:(-\infty,\infty) \rightarrow (0,1), f(x)=\frac{1}{2} \left (1+\frac{2}{\pi} \tan^{-1} x\right), f'(x)=\frac{1}{\pi (1+x^2)}>0.$$
Yet another example is 
$$f(x)= e^x,~ \mbox{if}~x<0;~ f(x)=2-e^{-x},~\mbox{if}~ x>0.$$
