Determining Convergence/Divergence of a Sequence

I am given a series defined by the following statement:

$a_n = $ the first digit in the decimal expression $n$

And so the sequence looks like this:

$1,2,3,4,5,6,7,8,9,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3,3,3$ etc.

I need to determine whether this sequence converges or diverges. My first instinct is simply to state that it is clearly infinite and non-repeating and so diverges. However I feel like there's a way to prove that is the case but can't figure out how since there's no general expression (that I can think of) for the sequence whose limit I can take.

Answer

for $n\geq1$, we have

$a_{10^n}=1$ and $a_{2.(10^n)}=2$.

two subsequences have two different limits, so the sequence $(a_n)$ is divergent.