MHB Solving a Challenging Number Sequence Problem

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The problem presented involves a sequence of positive integers where the n-th decimal digit has only finitely many zeros. The challenge is to prove that there are infinitely many positive integers that do not appear in this sequence. The discussion invites participants to engage with the problem and explore potential solutions. The poster expresses hope that others will enjoy tackling this mathematical challenge. The thread aims to foster collaboration and problem-solving within the forum community.
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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
 
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Well here you got the solution.

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.
 
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