Proving that cardinality of the reals = cardinality of $[0,1]$

$$ \begin{array}{rcl} [0,1/2) & \longleftrightarrow & (0,1] \\ [1/2,3/4) & \longleftrightarrow & (1,2] \\ [3/4,7/8) & \longleftrightarrow & (2,3] \\ [7/8,15/16) & \longleftrightarrow & (3,4] \\ \vdots & \vdots & \vdots \end{array} $$ This gives you a bijection between $[0,1)$ and $(0,\infty)$. Then $\log$ gives you a bijection between that and $\mathbb R$.

The correspondence $x\leftrightarrow \tan(\pi x + \pi/2)$ is a bijection between $(0,1)$ and $\mathbb R$. If you've already shown that $[0,1]$ and $[0,1)$ have the same cardinality, you should be able to show that $(0,1)$ and $[0,1]$ do, too.