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: Consider $\phi : \mathbb R \to (-1,1)$ defined by $$\phi(x) = \frac{x}{1  +|x|}$$ and notice that $f: (-1,1) \to (a,b)$ defined by $$f(x) = \frac{1}{2}\bigg((b-a)x + a+ b\bigg)$$
 is a bijection.
