I'm currently reading the Elliptic Curves Number Theory and Cryptography.
There is a proof on page 164. As far as I understand, it uses p-adic notation. I'm not very confident with the p-adics yet, so I've been confused by this proof for a while.
First, they introduced this rule:
$$ \frac{a}{b + O(p^k)} = \frac{a}{b} + O(p^k),\text{when}\ v_p(b) = 0, v_p(a) \ge 0, k > 0 $$
Then in the proof they derived the coordinates of the point on elliptic curve:
$$ x = \frac{m_1^2}{p^2} + O(p^{-1}) \\ y = -\frac{m_1^3}{p^3} + O(p^{-2}) $$
which I kind of understood.
And then they did this:
Since there's no explanation I tried to derive this result by myself:
$$ \frac{x}{y} = \frac{\frac{m_1^2}{p^2} + O(p^{-1})}{-\frac{m_1^3}{p^3} + O(p^{-2}} = \frac{m_1^2 * p + O(p^2)}{-m_1^3 + O(p)} = $$
$$ \frac{m_1^2 * p + O(p^2)}{-m_1^3} + O(p) = O(p) $$
I can't understand where's my mistake in reasoning, since I used the rule above properly:
$$ v_p(m_1^3 * p + O(p^2)) = 1 \ge 0, v_p(-m_1^3) = 0, k = 1 \gt 0 $$