I was wondering about how we could prove the completeness of $\mathbb R$ when this set is defined to be the set of all decimal expressions of the form : $$\underbrace{-}_{\text{sign}}\underbrace{317}_{\text{finite sequence of digits}}. \underbrace{12554382...}_{\text{possibly infinite sequence of digits}}$$ By the completeness of $\mathbb R$, I mean any property equivalent to the completeness axiom in a theory of the real numbers : it informally states that $\mathbb R$ has "no gaps".
Let's restrict ourselves to the interval $[0,1]$. With that in mind, let's define the set $D$ of all possibly infinite sequences of digits. Of course, $D$ is the set of all possible decimal parts of a real number in decimal expansion, and $D$ is essentially in bijection with $[0,1]$ (once we identify expressions like $5$ and $4999...$, because $0.5 = 0.4999...$).
By Cantor's diagonal argument, we already know that such set of sequences of digits is uncountable. But uncountability isn't completeness yet.
So is there a way to prove the completeness of $D$ ?