## 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 $$\mathcal O_{X,x}$$.