I have a previous similar question. I'm working out that one with the answerer, but I'm trying to gain insight from a different angle, especially in approaching these problems.
I must establish that two vectors $A$ and $B$ are equal asymptotically (length of vectors, $n\to\infty$). I consider the error vector, $e=A-B$ and try to show that $e\to 0$ as $n\to\infty$.
I then show that for each element $e_i$ of the error vector, $\Vert e_i\Vert_2\to 0 $ as $n\to\infty$. How can I proceed from here? Is this sufficient to say that the two vectors are equal in some sense?