/sci/ - Science & Math

Is it just me or is the proof given in https://en.wikipedia.org/wiki/Ostrowski%27s_theorem flawed?
in particular, they claim that for some integers [math]a,b\gt 1[/math] and some norm, which I will just label
[math]\left|\cdot\right|_\mathcal{A}[/math] on [math]\mathbb{Q}[/math] we get the following if expanding [math]b^n[/math] in base a:
[eqn]\left|b^n\right|_\mathcal{A} = \left|\sum_{j\lt m} c_j a^j\right|_\mathcal{A}\le a m \max(1,\left|a\right|_\mathcal{A}^{m-1})[/eqn]
however, the way i see it:
[eqn]\left|b^n\right|_\mathcal{A} = \left|\sum_{j\lt m} c_j a^j\right|_\mathcal{A}\le \sum_{j\lt m}\left| c_j \right|_\mathcal{A}\left|a\right|_\mathcal{A}^j\le m\left| a-1 \right|_\mathcal{A}\max(\left|a\right|_\mathcal{A}^{m-1},1)[/eqn]
because [math]c_j \lt a[/math] ! You can then continue [math]\left| a-1 \right|_\mathcal{A}\le a-1[/math]
Of course, that error does not change much of the proof because while you would have to correctly state
[eqn]\left|b\right|_\mathcal{A} \lt \sqrt[n]{ ma\max(\left|a\right|_\mathcal{A}^{m-1},1)}[/eqn]
Which, given [math] m\le n\log_a{(b)}+1[/math], would continue to yield for [math]n\rightarrow\infty[/math]
[eqn]\left|b\right|_\mathcal{A} \le\lim_{n\rightarrow\infty} \sqrt[n]{ (n\log_a{(b)}+1)a\max(\left|a\right|_\mathcal{A}^{m-1},1)} = \max(\left|a\right|_\mathcal{A}^{m-1},1)[/eqn]
But still, isn't it an error?