I was reading the article about $p$-adic integers from Wikipedia. The algebraic approach described there seems very interesting to me.
So far I understand that if $p$ is a prime then every natural number $m$ defines a sequence $(a_n)$ by $a_n = m \bmod p^n$ and this sequence defines a $p$-adic integer. Such a sequence eventually stabilises like $(1, 3, 3, 3, 3, 35, 35, 35, …)$. On the other hand an if we have a $p$-adic integer given by an ever-increasing sequence like $(1, 3, 7, 7, 23, 55, 55, 183, 439, 439, 1463 ...)$ then that corresponds to no natural number. But it might be possible that this sequence is corresponding to a real number.
So my question is
If a prime $p$ and an ever-increasing sequence $(a_n)$ is given then is it possible to determine which real number it corresponds to?
In particular if I have some formula involving $n$ and $p$ for $a_n$, say $a_n=2^n-2$ with $p=2$, then is there any way to determine/guess which real number $(a_n)$ corresponds to?
Any help will be greatly appreciated.