Normal is actually a stronger claim than "contains any finite string as a substring". That normal numbers contain any finite string as a substring is a straight

Normal is actually a stronger claim than "contains any finite string as a substring". That normal numbers contain any finite string as a substring is a straightforward consequence of the infinite monkey theorem: https://en.wikipedia.org/wiki/Infinite_monkey_theorem https://en.wikipedia.org/wiki/Infinite_monkey_theorem To see that the converse does not always hold, you could take something like the Champernowne constant https://en.m.wikipedia.org/wiki/Champernowne_constant https://en.m.wikipedia.org/wiki/Champernowne_constant and pad it with 9s between each integer, ie .192939495969798999109911991299... so that you still contain every finite substring, but you have a >50% chance of a randomly selected digit being 9.