Prove: Any open interval has the same cardinality of $\Bbb R$ (without using trigonometric functions) I want to prove that every open interval has the same cardinality of $\Bbb R$.
The question is:
Is it enough to prove that any open interval is uncountable? If I prove it, can I say that this interval has the same cardinality of reals?
I know that I can get a bijection using the tangent function, but I am not allowed to use trigonometric functions.
I've proved that $|(a,b)|=|(0,1)|$ so I may prove that $|(0,1)|=|\Bbb R|$ but I did not find the bijection without using trigonometric functions.

Consider the function
$$g(x)=\frac{x}{1+|x|}$$
Verify that $g$ is a bijection from real numbers to $(-1,1)$.
