In Principles of Mathematical Analysis(Rudin) , 2.14 themorem said A be the set of all sequences whose elements are the digits 0 and 1, then A is uncountable.
I understand the proof of these theorem.
My question is that we can represent all natural number to binary number such like 4 = 100. In that case we can consider the natural numbers as the set of sequence whose elements are the digits 0 and 1. By doing so, natural numbers are not countable sets.
What's the point I missed??
"Sequence" here refers to an infinite string of $0$s and $1$s, also known as an infinite binary string.
Each natural number has a finite representation as a binary string.
What's uncountable is the set of all infinite binary strings.