Let $u_1,u_2 \in \mathbb R^n$ be two vectors in an Euclidean space. Suppose that for every integer $k \in \mathbb Z$ we have $|u_1| \le |u_2+ku_1|$. I need to show that for every $k_1, k_2 \in \mathbb Z$, (not both $0$) we have: $|u_1| \le |k_1u_1+k_2u_2|$.
I used the triangle inequality, inserted the $k_2$ in the equation but couldn't get back from a sum of norms to a norm of a sum as asked, because the triangle inequality goes one way if you know what I mean :) Also, the Cauchy-Schwarz inequality is not useful (I believe) here since we have to do with the vectors themselves and not the coordinates (but I may be worng).
If this needs a trick to be solved, can anyone provide a hint?