Finite vs Infinite Math Example 2

Solution

1. 1
   Between $0$ and $1$ there are infinitely many decimals (e.g., $0.1, 0.01, 0.001, \ldots$ is already an infinite list). So the set is infinite.
2. 2
   Cantor's diagonal argument shows this set is uncountable: suppose we list all decimals in $[0,1]$; we can always construct a new decimal that differs from the $n$-th listed decimal in the $n$-th decimal place, so the list was incomplete.
3. 3
   Therefore the decimals in $[0,1]$ form an uncountably infinite set, strictly 'larger' than the natural numbers.