I'm given a positive number, a unit vector $u \in \mathbb{R} ^n $ and a sequence of vectors $ \{ b_k \} _{ k \geq 1} $ such that $|b_k - ku| \leq d $ for every $ k=1,2,...$.
This obviously implies $ |b_k| \to \infty $ . But why does this imply $\angle (u,b_k) \to 0 $ ? I've tried proving it using some inner-product calculations, but without any success.
In addition, why the given data implies that there must exist $i
Thanks a lot !