Thanks for this nice problem!
By induction over :
- obvious if ( : the 0-dimensional vector space)
- take and assume this is true for : among any -uplet of vectors in (or in any -dimensional Euclidean vector space, equivalently), there are (at least) two vectors that have nonnegative dot product. Let . If any vector is zero, the result is obvious, so we may assume . Denote by the projection on the -dimensional subspace . By induction, there are such that . Then, consider . I claim that one of , or is nonnegative. Indeed, if and , then (because for any vector , ).