May as well go through the proofs:
First, we need to establish that two infinities are equal in cardinality (aka size) if all their elements can be 1:1 mapped to each other.
So, to go from the reals within [0, 1] and [0, 2], we can multiply by 2. This maps every value within [0, 1] to every value within [0, 2], so these are of the same cardinality.
Where things get interesting is the proof that the reals within [0, 1] are of greater cardinality than every integer.
Say we have an arbitrary mapping from every integer to a real within [0, 1]:
0 -> 0.89236…
1 -> 0.47389…
2 -> 0.84776…
3 -> 0.18790…
4 -> 0.90542…
⋮ ⋱
This list contains every integer, but it does not contain every real number because we can always come up with a new one by ensuring at least one digit is different in each existing real:
0 -> …8… ≠ 9
1 -> …7… ≠ 8
2 -> …7… ≠ 8
3 -> …9… ≠ 0
4 -> …2… ≠ 3
⋮ ⋱
0.98803… is not within the list
Therefore, no 1:1 mapping between the integers and reals exists. Because the limiting factor is the amount of integers, the cardinality of the reals is greater than that of the integers.
Edit: https://en.wikipedia.org/wiki/Cantor's_diagonal_argument