So I know the rationals are countable, but the reals aren't countable.. I don't understand this as I thought.
How do I show that the reals are uncountable?
Did you want a proof to show the rationals are countable too? I'm not sure exactly what you're wanting.
Theorem: $\displaystyle \mathbb{R}$ is uncountable.
Proof: Suppose $\displaystyle \mathbb{R}$ is countable.
We can enumerate the reals $\displaystyle x_1, x_2, x_3\ldots$
We'll create $\displaystyle x \in \mathbb{R}$ that is NOT in the list.
Pick the 10's decimal to differ from $\displaystyle x_1$
Pick the 2nd decimal place to differ from $\displaystyle x_2$
.
.
.
Example:
$\displaystyle x_1 = 17.3189; x_2 = -329.018726; x_3 = 0.0009119\dots$
The only decimals we have to be careful about are 9's and 0's (for instance if we had .99999... because that is equal to 1).
Thus, make sure to pick digits other than 9 and 0.
Thus, x is not in the list, which implies $\displaystyle \mathbb{R}$ is uncountable. $\displaystyle \square$