Suppose $K'/K$ is a totally ramified extension of $p$-adic fields of degree $e.$ A paper (p.9, line 15) I am reading seems to use the following formula for the upper numbering on the absolute galois groups $$G_{K'}^{eu}=G_{K}^{u}$$ for $u>0.$
They quote Proposition 15, Chapter IV of Serre's Local Fields for this fact, which says that the Herbrand function $\varphi_{F/L}$ and its inverse $\psi_{F/L}$ satisfy transitivity formulas $$\varphi_{F/L}=\varphi_{L'/L}\circ \varphi_{F/L'}\text{ and }\psi_{F/L}=\psi_{F/L'}\circ\psi_{L'/F}$$ for an extension of fields $F/ L' / L.$
How does the formula above follow from this proposition? Or is the formula not true in general (quite possibly I am missing some subtlety that is making this formula work in the paper)?
The extension $K'\supset K$ is tamely ramified in the paper you’re reading. But in the case of tame ramification, everything is concentrated at the origin of the Herbrand graph. That is, $\varphi^{K'}_K(x)=x/e$ for $x\ge0$. All follows from that.