Embedding of valued fields I am reading this Lecture notes on the model theory of valued fields. On p.71, he states the following lemma:

6.12. Lemma. If $(K, v, \Gamma)$ is $\aleph_{1}$ -saturated, then the discrete valued field $(\dot{k}, v, \mathbb{Z} 1)$ is complete.

which is left as an exercise. I cannot prove it on my own.
My attempt:
I know that $\mathcal{M}$ is saturated if every type over a subset of smaller cardinality than M is realized. But I do not know how to actually use that.
I would apprecaite any help or reference to a paper or a book where this lemma is proven.
 A: 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.
