so I'm a bit confused about convergence over the $p$-adic numbers - for example, I would argue that $\frac{1}{5^n}$ is not convergent over $\mathbb{Q}_5$ since $|\frac{1}{5^n}|_p = 5^n$ which is not convergent over $\mathbb{R}$, but I feel I may have switched to the reals too early? I'm not sure about the intuition - are $n$, $n!$ convergent over $\mathbb{Q}_5$?
Thanks!
On
You're correct; $p^n$ diverges as $n\to-\infty$ and converges to $0$ as $n\to+\infty$ in $\Bbb Q_p$.
For a sequence of integers to converge in $\Bbb Q_p$, a necessary condition is that the $p$-adic valuations of the terms must eventually stabilize to some constant sequence or tend to $+\infty$. Can you argue why this must be true? What can you say about the exponents of $p$ within $n$ or $n!\,$?
(Also, fun fact: last I heard the ir/rationality of $\sum\limits_{n=1}^\infty n!$ is not known in $\Bbb Q_p$ for any $p$.)
Being small is being divisible by a high power of $p$. So indeed $\frac1{p^n}$ looks small to the "real" eye, but is $p$-adically big - the sequence $(\frac1{5^n})$ does not converge in the 5-adics.
The sequence $(n)$ does not convereg. In contains small numbers (such as $3125$) evera now and then, but also non-multiples of $5$ (i.e. medium-sized numbers, or as big as they can for integers).
The sequence $(n!)$ finally converges to $0$ because the terms become smaller and smaller (more and more factors of $5$)