I'm reading Henning Stichtenoth's Algebraic Function Fields and Codes and at Proposition 3.2.5(a) of section 2, chapter 3 he says:
Let $\mathcal{O}_S$ be a holomorphy ring of $F/K$. Then $F$ is the quotient filed of $\mathcal{O}_S$.
At the proof he takes an element $0\neq x\in F$ and a place $P_0\in S$. Then he says that by the Strong Approximation Theorem there exists an element $z\in F$ such that (the function $v_P:F\to\mathbb{Z}\cup\{\infty\}$ is the valuation function associated with the given place $P$):
$$v_{P_0}(z)=\text{max}\{0,v_P(x^{-1})\}\quad\text{and}\quad v_P(z)\geq\text{max}\{0,v_P(x^{-1})\}\;\forall P\in S\setminus\{P_0\}$$
I can't see why the inequality holds. For you to understand completely I'll enunciate the Strong Approximation Theorem (Theorem 1.6.5 of the book):
Let $S\subset\mathbb{P}_F$ be a proper subset of $\mathbb{P}_F$ and $P_1,\ldots,P_r\in S$. Suppose there are given elements $x_1,\ldots,x_r\in F$ and integers $n_1,\ldots,n_r\in\mathbb{Z}$. Then there exists an element $x\in F$ such that $$v_{P_i}(x_i)=n_i,\;i=1,\ldots,r\;\text{and}\;v_P(x)\geq0\;\forall P\in S\setminus\{P_1,\ldots,P_r\}$$
Back to Proposition 3.2.5(a), I tried to consider, by the Strong Approximation Theorem, an $\alpha\in F$ such that
$$v_{P_0}(\alpha - x)=\text{max}\{0,v_{P_0}(x)\}\quad\text{and}\quad v_P(\alpha)\geq 0\;\forall P\in S\setminus\{P_0\}$$
and then take $z=\alpha-x$, but that doesn't help so much because, by the Triangle Inequality: $$v_P(z)=v_P(\alpha-x)\geq\text{min}\{v_P(\alpha),v_P(-x)\}=\text{min}\{v_P(\alpha),v_P(x)\}$$
So why does that first inequality follow from the Strong Approximation Theorem?
Let $\lbrace P_1,\dotsc,P_r\rbrace$ be the places in $S$ where $v_{P_i}(x^{-1})>0$ (there are only finitely many.) Then the inequalities you wrote are satisfied by any $z$ with $$v_{P_0}(z)=\max\lbrace 0,v_{P_0}(x^{-1})\rbrace$$ and $$v_{P_i}(z)=v_{P_i}(x^{-1}),\,\,i=1,\dotsc,r$$ and $$v_P(z)\geq 0\,\,\textrm{if}\,\,P\in S\backslash\lbrace P_0,P_1,\dotsc,P_r\rbrace.$$ Such a $z$ exists by the strong approximation theorem: The first two inequalities above are from the first inequality in the strong approximation theorem (take the $x_i$'s in that theorem to be zero), and the third inequality above is from the second inequality in the strong approximation theorem.