elementary set theory - Countability of Infinite Sets

Show that $|\mathbb{R}|$=$|[0,1]|$.

If we were to find a function whose domain is $\mathbb{R}$ and range is $[0,1]$ and show it is a bijection, then we can show that this is true. The function that I came up with is $f(x)=\frac{\arctan x+\frac{\pi}{2}}{\pi}$, but this function's range is $(0,1)$. Is there a function that would work?