Approximating a $2$-adic integer by an odd integer

Question

I am looking at a Lemma in number theory and think I have found a proof but it requires a result that I'm not sure is true in general. If we have $x \in \mathbb{Z}_2$ can we find an odd $x_0 \in \mathbb{Z}$ such that $|x-x_0|_2$ can be arbitrarily small?

Thanks.

Answer 1

For any odd number $x_0\in\Bbb Z$, we have $\left|x_0\right|_2 = 1$, as $2$ occurs to the $0$th power in the prime factorization of $x_0$. By the nonarchimedean triangle inequality, $\left|x - x_0\right|_2\leq\max\{\left|x\right|_2,\left|x_0\right|_2\}$, with equality if $\left|x\right|_2\neq\left|x_0\right|_2$. Thus, if your given $x$ does not satisfy $\left|x\right|_2 = 1$, $\left|x - x_0\right|_2 = 1$ for any odd $x_0\in\Bbb Z$, so that $\left|x - x_0\right|_2$ cannot be arbitrarily small.