Let $(v\_i)\_{i∈ℕ}$ be a vector sequence. Say $(v\_i)$ is *boundable (under $M$)* if there exists a sequence $(b\_i)\_{i∈ℕ}$ taking values in $\{-1,1\}$ such that $(|\sum^N\_i b\_iv\_i|)\_{N∈ℕ}$ is bounded (under $M$). If $(v\_i)$ takes values in the unit ball $B^n⊆ℝ^n$, does it follow $(v\_i)$ is boundable?

With the definition formulated analogously for finite sequences, I see I can utilize the (weak) Konig's lemma to show that a vector sequence is boundable under $M$ if and only if all its finite starting sequences are boundable under $M$. As such, it follows that if every such vector sequence is boundable, then there is a minimal such $α\_n$ such that every vector sequence is boundable under $α\_n$ (for example, $α\_1=1$). Based on some randomized computations of mine it appears the hypothesis holds for $n=2$ and $1.6≤α\_2≤1.7$, however I'm not sure how to continue.

**Edit:** It appears actually that should the hypothesis be true, then $α\_2 ≥ \sqrt{3}$. [This](https://i.imgur.com/CVmsbgA.png?1) diagram indicates how to construct a finite sequence unboundable under $r$ when $r < \sqrt{3}$: Start with the resultant vector $w\_1$ (initial $v\_1=(1,0)$) of magnitude $1$ or greater; choose the next vector $w\_2$ such that $|w\_1+w\_2| > r$ yet the angle between $w\_1,w\_2$ is less than $2π/3$. Then $|w\_1-w\_2| > |w\_1|$ and we can repeat the process with resultant vector $w\_1-w\_2$ to at least constantly greater effect each time. (Note the mirror situation of $b\_1=-1$ is taken care of by symmetry)