Let $K$ be a complete, algebraically closed non-archimedean field. The Arzela-Ascoli theorem fails over such a field $K$.
I found an example on a book, but not quite get it. Here is an example :
$\color{blue}{Let\,\,\, \{x_n\} \,\,be \,\,a\,\, sequence \,\,of\,\, points\,\, in\,\, \mathbb{P}^1(K)\,\, with\,\, no\,\, accumulation\,\, points.\,\, Thus\,\, the\,\, family\,\, of\,\, constant\,\, functions\,\, \{x_n: n\geq 0\}\,\, is\,\, equicontinuous\,\, but \,\,not\,\, normal\,\, on\,\, any\,\, open\,\, subset\,\, U\subseteq \mathbb{P}^1(K).}$
It would be good if someone could explain this to me.
By any chance, is it possible to write such sequence $\{x_n\}\subset\mathbb{P}^1(K)$ concretely?
So the normality and equicontinuity of family of functions cannot be used interchangeably over non-archimedean field, right?
We need that $K$ has an infinite residue field, it doesn't work with $K=\Bbb{Q}_p$.
(For $p\ne 2$) with $K= \Bbb{Q}_p(\zeta_{2^\infty})$ (well its completion in fact) take $$x_n=[\zeta_{2^n}:1]$$ $$d(x_n,x_m)=\inf_j |x_n(j)-x_m(j)|_p=1$$