I was working on the Introduction to Number theory textbook by CJ Bradley and encountered the following theorem .
If $N>0$ and $\frac{u}{v}$ is a good rational approximation to $\mathrm{N}^{\frac{1}{p}}$ , where $p$ is a positive integer, then $$ \frac{U}{V}=\frac{u\left\{(p-1) u^{p}+(p+1) v^{p} N\right\}}{v\left\{(p+1) u^{p}+(p-1) v^{p} N\right\}} $$ is a far better approximation, in the sense that if $$ \left(\frac{u}{v}\right)^{p}=N\left(1+\frac{\varepsilon}{N}\right) \text { and }\left|\frac{\varepsilon}{N}\right|<1, $$ then $$ \left(\frac{U}{V}\right)^{p}=N\left(1+\mathcal{O}\left(\frac{\varepsilon^{3}}{N^{3}}\right)\right) . $$ Note that $\mathcal{O}$ means the order of the given expression
The book states that the proof is straightforward and requires only the use of the binomial theorem. Naturally, I tried applying it but i can't seem to prove it. A direction to work upon or a proof will suffice and will be appreciated Thanks a lot