Reference wanted for generalized Dirichlet's Theorem

I'm looking for a reference for the following result, which is a generalization of the classical theorem of Dirichlet on the approximability of real irrationals by rational numbers:

Let $k$ be a number field, $O$ its ring of integers, $v$ an infinite place of $k$, $\alpha$ any element of the completion $k_v$. Let $\|\cdot\|_v$ be the usual absolute value (or its square, if $v$ is a complex place). Let $H$ denote the multiplicative height function relative to $k$ -- that is, for any element $x\in k$, let $H(x)=\prod_w \max(1,\|x\|_w)$, where the product is over all places $w$ of $k$. Then there is a positive real constant $C$ depending only on $k$ such that

$$\|\alpha-x\|_v < \frac{C}{H(x)^2}$$

for infinitely many $x\in k$.

I think I can prove this, but I am surely not the first. If anyone can tell me a good place to point to for this result, I'd be very grateful -- thanks!

• For $k=Q$ the statement is false: a non-integral $\alpha$ cannot be approximated well with integral $x$. Dirichlet's Theorem is concerned with approximation by rational numbers, not integers. – GH from MO May 7 '12 at 18:35
• I changed the $O$ to $k$. – Joe Silverman May 7 '12 at 19:02
• Then $O$ is not needed in the third line either. – GH from MO May 7 '12 at 20:32