Take a look at 0.101001000100001… This number is also non-repeating, but obviously doesn’t contain all numbers with finite digits.

The property you’re looking for is called to be a normal number. Pi is assumed to be one, but it hasn’t yet been proven.

However, in a sense this is an unremarkable property as almost all real numbers are normal. :)

Your fact is correct, but the mind-blowing thing about infinite sets is that they go against intuition.

Even if one might think that the number of odd numbers is strictly less than the number of all natural numbers, these two sets are in fact of the same size. With the mapping n |-> 2*n - 1 you can map each natural number to a different odd number and you get every odd number with this (such a function is called a bijection), so the sets are per definition of the same size.

To get really different “infinities”, compare the natural numbers to the real numbers. Here you can’t create a map which gets you all real numbers, so there are “more of them”.