Embedding of valued fields

If you made it to Proposition 4.21, then you can proceed as follows. Use Zorn's lemma on the set of (valued) subfields $L$ with $K\subseteq L\subseteq K^*$, together with embeddings $L \rightarrow K'$ over $K$, ordered by inclusion. Given a maximal element $(L,\varphi)$ of this set, the goal is to prove that $L=K^*$.

By the functorial property of the henselian closure (also called henselisation), we can assume that $L$ is henselian. Assume for contradiction that there is an $x \in K^* \setminus L$. If $x$ has algebraic type over $L$, i.e. $x$ is a pseudo-limit of a pseudo-Cauchy sequence of algebraic type, then by Proposition 4.21 and Theorem 4.10, one can etend $\varphi$ into an embedding $L(x)\rightarrow K'$. So we may assume that $x$ has transcendal type over $L$, i.e. there is a pseudo-Cauchy sequence $(a_{\rho})_{\rho <\lambda}$ in $L$ that has $x$ has a pseudo-limit. Theorem 4.9 states that for any pseudo-limit $y$ of $(\varphi(a_{\rho}))_{\rho <\lambda}$ in $K'$, we can extend $\varphi$ into an embedding $L(x) \rightarrow \varphi(L)(y) \subseteq K'$. So we need only show that $(\varphi(a_{\rho}))_{\rho <\lambda}$ has a pseudo-limit in $K'$.

In order to do this, choose $(a_{\rho})_{\rho <\lambda}$ such that $(v(x-a_{\rho}))_{\rho <\lambda}$ is strictly increasing (I let you figure this out, this should be an easy consequence of the lemmas in the beginning). In particular, this implies that $|\lambda|\leq|\Gamma|$. So the saturation hypothesis lets us pick a $y \in K'$ with $v(y-\varphi(a_{\rho}))>v(y-\varphi(a_{\gamma}))$ whenever $\gamma<\rho<\lambda$. In other words, $y$ is the desired pseudo-limit.