Do all natural numbers appear in the digits of a normal number?

Question

A normal number is a number where no number is favored to appear in the digits. Does this definition imply that all whole numbers appear in its digits? Because the definition involves notions from probability, I was wondering if it might happen that a certain number would not appear in the digits of a normal number without contradicting the definition.

Answer 1

Yes it does. Working in base B, a string with A digits should have the natural density $ \frac {1}{ B^A} $ by definition of a normal number. If the string doesn't appear at all, then it has natural density 0.