Show that all the intervals in $\mathbb{R}$ are uncountable 
Question:
Show that all the intervals in $\mathbb{R}$ are uncountable.

I have already proven that $\mathbb{R}$ is uncountable by using the following:
Suppose $\mathbb{R}$ is countable. Then every infinite subset of $\mathbb{R}$ is also countable. This contradicts the fact that $\mathbb{I} \subset \mathbb{R}$ is uncountable. Consequently, $\mathbb{R}$ must be uncountable.
However, how can I show that ALL the intervals in $\mathbb{R}$ are uncountable?

HINT: Find an injection from $\Bbb R$ into the interval $(0,1)$. Then show there is a bijection between any two intervals $(a,b)$ and $(c,d)$.
