I am currently reading Robert's A Course in p-adic Analysis textbook and having some trouble understanding some steps in his proof of the Mahler Theorem. At about the halfway point in the proof (top of p174), he uses the "competition principle" (which is never defined, only used) to state that at least one of the $a_k^0$'s have $|a_k^0| = 1$. Since I can't embed images yet, here is a link to the relevant parts of the proof: https://i.stack.imgur.com/H3CfJ.png
I tried googling it (and only found stuff about ecology) and looking at previous mentions in the textbook, but haven't found anything helpful. I would be very grateful if someone could explain what this principle is and possibly prove whatever it is stated as.
I guess by "competition principle" he means what is often called the "maximum principle" of ultrametric norms, namely that $\Vert x + y \Vert = \max(\Vert x \Vert, \Vert y \Vert)$ if $\Vert x \Vert \neq \Vert y \Vert$. In this case and with his notation, one would use the easy corollary that if $\Vert f \Vert =1$ but all of the $\Vert a_k^0 f_k \Vert < 1$ then $\Vert f - \sum a_k^0 f_k \Vert = 1$, which contradicts the image of that function being in $pA_p$.