Given $v_i∈B^n$, bounding $\sum b_iv_i$ for $b_i= \pm 1$ 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 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)

This answer is based on my misuderstanding of the question, see the second Feryll's comment below.

According to Wojciech Banaszczyk [B1], S. Sevastyanov [S] and, independently I. Bárány (unpublished) proved that $\alpha_n=\sqrt{n}$. Also there is a lot of the generalizations of this inequality for different norms, see, for instance, the references.
References 
[B1] Wojciech Banaszczyk, A Beck-Fiala-type Theorem for Euclidean Norms, Europ. J . Combinatorics 11 (1990), 497-500.
[B2] Wojciech Banaszczyk, Balancing vectors and convex bodies, Studia mathematica, 106 (1993), 93-100.
[G] Apostolos A. Giannopoulos,  On some vector balancing problems, Studia mathematica, 122:3 (1997), 225-234.
[S] S. V. Sevastyanov, On the approximate solution of the problem of calendar planning, Upravlyaemye Systemy, 20 (1980), 49–63 (in Russian). 
