I am working with Sam Payne's artical Analytification is the limit of all tropicalizations, and because of my limited understanding of analytification it gives me some difficulties.
The article: http://intlpress.com/site/pub/files/_fulltext/journals/mrl/2009/0016/0003/MRL-2009-0016-0003-a013.pdf
Let $K$ be a field with a valuation map. We consider a subvariety $X$ in the torus $T^n =(K^*)^n$. In the proof of Proposition 2.1 in the article he uses that $X(K)$ is a subset of $X^\text{an}$. I am very interested in understanding why we can consider $X(K)$ as such a subset. I have tried to prove this. Thus I want to show that every point $p \in X(K)$ induces a norm on $X^\text{an}$. This however has given me a lot of difficulties. Anyone who can give an argument (rigorous or not) to why we can consider $X(K)$ as a subset of $X^{\text{an}}$.