Some intuition for uncountability:
countably many points will never fill up a line!
This can be made precise,
by covering each point with an interval of length
$\epsilon 2^{-n}$:
then they overall cover $\epsilon$,
so there's no way they cover the whole line.
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BTW, Cantor's 1st proof of uncountability of the reals
is a good exercise for students.
Give them the set-up
"Assume axioms,
assume an enumeration,
now define $a_n$ and $b_n$ in this way
(this kind of double sequence dates back to the Greeks!)
...and derive a contradiction"