Solving a Challenging Number Sequence Problem

[Fallen Angel]

Hi!

I'm new in the forum and I bring a challenge

Here is the problem:

For a positive integer $$x$$, denote its n-th decimal digit as $$d_{n}(x)$$, with $$d_{n}(x)\in \{0,1,\ldots ,9\}$$ so $$x=\displaystyle\sum_{i=1}^{+\infty}d_{i}(x)10^{i-1}$$.

Let $$(a_{n})_{n\in \Bbb{N}}$$ be a squence such that there are only finitely many zeros in the sequence $$(d_{n}(a_{n}))_{n\in \Bbb{N}}$$.

Prove that there are infinitely many positive integers that do not occur in the sequence $$(a_{n})_{n\in\Bbb{N}}$$.Hope you enjoy it! :p

 

[Fallen Angel]

Well here you got the solution.

Spoiler

Let $$n_{0}=max\{n\in \mathbb{N} \ : \ (d_{n}(a_{n}))=0$$
Until this point there are just $$n_{0}$$ positive integers that occur in the sequence $$(a_{n})$$.
Then we got that for all $$n > n_{0},\ a_{n}\geq 10^{n-1}$$
Hence , there are $$10^{n-1}-n$$ positive integers that can not occur in the sequence $$(a_{n})$$ till the n-th position.
Taking the limit we got the result.