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?
Let $\pi$ be the orthogonal projection along $u_1$. Then there is a $k \in \mathbb{Z}$ such that $\lVert (1-\pi)(u_2+ k u_1) \rVert = \lVert (1-\pi) u_2 + k u_1 \rVert \leq \tfrac12 \lVert u_1 \rVert$. For this $k$ we have $$\begin{eqnarray}\lVert u_1 \rVert^2 &\leq & \lVert u_2+ k u_1 \rVert^2 \\ &= & \lVert (1-\pi)(u_2+k u_1)\rVert^2 + \lVert \pi(u_2 + k u_1)\rVert^2 \\ &\leq & \tfrac14 \lVert u_1 \rVert^2 + \lVert \pi u_2 \rVert^2 \end{eqnarray}$$ and so $\lVert \pi u_2 \rVert > \tfrac12 \lVert u_1 \rVert$. Then for $k_1, k_2 \in \mathbb{Z}$ and $\lvert k_2 \rvert \geq 2$ $$\lVert k_2 u_2 + k_1 u_1\rVert \geq \lVert \pi(k_2 u_2 + k_1 u_1) \rVert \geq 2 \lVert \pi u_2 \rVert > \lVert u_1 \rVert.$$ The cases with $\lvert k_2 \rvert <2$ are easy to check.