I need help solving this number theory exercise related with best aproximation by continued fractions. Im using BROCHERO, F., MOREIRA, C.G., SALDANHA, N., TENGAN, E. – Teoria dos números – um passeio pelo mundo inteiro com primos e outros números familiares, Projeto Euclides, IMPA, 2010.
Let $\alpha=\left[a_0 ; a_1, a_2, \ldots\right] \in \mathbb{R} \backslash \mathbb{Q}$. If $\sqrt{5}<k(\alpha)<29 / 10$, show that there exists $N \in \mathbb{N}$ such that $a_n=2$ for all $n \geq N$, and conclude that $k(\alpha)=2 \sqrt{2}$.
Edit: $k(\alpha) = \sup \left\{k>0 \, \middle| \, \text{there exist infinitely many rational solutions } \frac{p}{q} \text{ satisfying } \left|\alpha-\frac{p}{q}\right| < \frac{1}{k q^2}\right\} \\ = \limsup_{p, q \in \mathbb{Z}, q \rightarrow \infty} \left(|q(q \alpha-p)|^{-1}\right) \in \mathbb{R} \cup\{+\infty\}$.
We can derive a formula for $k(\alpha)$: we write $\alpha$ in continued fraction form, $\alpha=\left[a_0, a_1, a_2, \ldots\right]$, and define, for $n \in \mathbb{N}$, $\alpha_n=\left[a_n, a_{n+1}, a_{n+2}, \ldots\right]$ and $\beta_n=\left[0, a_{n-1}, a_{n-2}, \ldots, a_1\right]$. Then we have $k(\alpha)=\limsup_{n \rightarrow \infty}\left(\alpha_n+\beta_n\right)$. In particular, $k(\alpha)<\infty \Longleftrightarrow$ $\left(a_n\right)$ is bounded.