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

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

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

 

[Fallen Angel]

Well here you got the solution.

Spoiler

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