I am trying to read a proof of the result that given a field $F$ that is complete with respect to a discrete valuation $v$, and an extension $K/F$ of dimension $n$, that there exists a unique discrete valuation $v_K$ on $K$ extending $v$, which is given by $v_K(\alpha)=(1/n)v(det(m_{\alpha}))$. In particular, I am having trouble with the following statement at the beginning of the proof: "We must prove that $v_K(\alpha+\beta)\geq min(v_K(\alpha),v_K(\beta))$ for all $\alpha,\beta$, and that is clearly equivalent to proving that $v_K(1+\alpha)\geq 0$ if $v_K(\alpha)\geq 0$ for all $\alpha$."
The first statement implying the second statement is clear, but how does the second imply the first? The wording is also weird. Is it that we must have $v_K(\alpha)\geq 0$ for all $\alpha$ in $K$, or that for all $v_K(\alpha)\geq 0$ the inequality must apply?
I do hope that this wasn’t in a published text. For, one must never never never put the quantifiers at the end of the statement.
The first statement should have been written, “$\forall\alpha,\beta\in K, v(\alpha+\beta)\ge\min\bigl(v(\alpha),v(\beta)\bigr) $”, and the intention of the author or instructor was probably, “$\forall\alpha\in K$ with $v(\alpha)\ge0$, we have $v(1+\alpha)\ge0$” .
Perhaps you will see now that the way to go from the second formulation to the first is to start with two elements $\gamma,\delta\in K$, say $v(\gamma)\ge v(\delta)$, and apply the second formulation to $\gamma/\delta$.