Let $K$ be a complete field with $\operatorname{char}(K) = 0$ and a non trivial ultrametric absolute value $||$, Such that $K$ is a metric space.
Let $z$ a transcendental element over $K$.
We define $K\{z\} = \left\{ \sum\limits_{n=0}^{\infty} {a_n z^n} \mid a_n \in K , {a_n} \rightarrow 0 \right\} $
Let $a \in K$ such that $|a| < 1$, and $f(z) \in K\{z\}$
How can one show that also $f(z+a) \in K\{z\}$ ?
Thanks in advance!