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