Why is $k\subseteq A/\mathfrak p\subseteq k(x)$?

The following lemma is from Qing Liu’s “Algebraic Geometry and Arithmetic Curves” p.61. In line 4 of the proof, why is $ k\subseteq A/\mathfrak p\subseteq k(x)$ ? I think it means that there are injective ring homomorphisms $ k\to A/\mathfrak p$ and $ A/\mathfrak p\to k(x)$ , but I don’t know how to construct such homomorphisms.

$ k(x)$ is the residue field $ \mathcal O_{X,x}/\mathfrak m_x$ , where $ \mathfrak m_x$ is the maximal ideal of $ \mathcal O_{X,x}$ .

Lemma 4.3