Units in a number field Let $K=\mathbb{Q}[\sqrt{229}]$ and let $z\in K$ be an element of norm $\pm3$. Prove that after possibly multiplying $z$ by a unit in $\mathbb{Z}_{K}^{\times}$ we may assume that $u^{-1/2}\leq z\leq u^{1/2}$ where $u$ is a fundamental unit.
So far I have shown that $u=7+\frac{1+\sqrt{229}}{2}$ is a fundamental unit and I have no idea what to do now.
 A: I suspect you're too focused on the details of the problem that you're missing the obvious. Try this exercise:

Let $x$ and $y$ be positive real numbers. Prove there exists an integer $n$ such that $x^{-1/2} \leq x^n y \leq x^{1/2}$

A: I think that some basic facts about units have not been properly explained to you, and that's after I read some comments giving a little more of the context.
I would like to use $\eta$ (Greek lowercase eta) to represent the fundamental unit, and $u$ to represent arbitrary units in the relevant domain, in this case $\mathcal O_{\mathbb Q(\sqrt{229})}$. If $u > 1$ and no smaller $u$ satisfies this strict inequality, then $u = \eta$. Yeah, I know, that's kind of the definition of the fundamental unit.
The fundamental unit is important because $\eta^n$ and $-(\eta^n)$, in which you iterate $n$ through all the integers of $\mathbb Z$, gives all the units of the domain. In particular, note that $\eta^0 = 1$, $-(\eta^0) = -1$ and $-\eta < -1$. So we have $-\eta < -1 < 1 < \eta$.
Also, even though $\lfloor | \eta^n | \rfloor$ can be quite large, we nevertheless have $N(\eta^n) = \pm 1$. So those are the basic facts to keep in mind here.
Now, you're looking to place $zu$ in a smaller range than between $-\eta$ and $\eta$, namely, $$\frac{1}{\sqrt \eta} < zu < \sqrt \eta.$$ That's roughly between 0.25762953 and 3.8815426, for this particular exercise in $\mathcal O_{\mathbb Q(\sqrt{229})}$. 
No number $z$ exists in this domain having a norm of 3 or $-3$, as Gerry Myerson explains in his answer to this question: Easiest way to show $x^2 - 229y^2 = 12$ has no solutions in integers
But if such a number $z$ existed, you could place $zu$ in the desired range by simple trial and error. If $zu$ is too large, try a smaller unit. And if it's too small, try a larger unit, secure in the knowledge that $N(zu) = \pm 3$ regardless.
In order to give a concrete example, let's say that instead of $N(zu) = \pm 3$ you're looking for $N(zu) = -509$. Then we can do $$z = \frac{5}{2} + \frac{3 \sqrt{229}}{2} = 1 + \frac{3 + 3 \sqrt{229}}{2} \approx 25.1991189.$$ Clearly $z \eta$ will be too large, and we needn't bother with $z \eta^0$. But $z \eta^{-1} = 153 - 10 \sqrt{229} \approx 1.67254$, putting $zu$ in the desired range.
