Let $A$ be a Dedekind domain, and let $K$ be its field of fractions. In Serre's Local Fields, the following Lemma is stated.
Approximation Lemma Let $k$ be a positive integer. For every $i$, $1\leq i \leq k$, let $\mathfrak p_i$ be prime ideals of $A$, $x_i$ elements of $K$, and $n_i$ integers. Then there exists an $x\in L$ such that $v_{\mathfrak p_i}(x - x_i)\geq n_i$ for all $i$, and $v_{\mathfrak q} \geq 0$ for $\mathfrak q \neq \mathfrak p_1,\ldots, \mathfrak p_k$.
To get a better feel for this I would like to be able to actually find the $x$ stated in Lemma. I have tried to follow the proof to do this, and there is one crucial step that I don't understand at all.
At the start, after assuming that the $x_i$ are in $A$, it is stated that "by linearity, one may assume that $x_2 = \ldots = x_k = 0$". I don't see why we can assume this, and I don't even know what is meant by linearity here.
Any suggestion as to what it means in the proof, or alternative (preferably constructive) proofs would be much appreciated.
As a further, related question, this lemma is often stated as showing that we can find an element that is in one collection of ideals, and not in some other ideal (for example, the proof of proposition 19). I don't see how the Lemma controls an element NOT being in an ideal - it has no control over how high the valuation at an ideal is. Any hints about this would be appreciated too.