For two unit vectors $u$ and $v$, prove that if $$\sin(\angle u,v)\leq l$$ We have: $$\exists \theta\in \lbrace -1,1\rbrace :\quad \|u-\theta v\|_2\leq\sqrt{2} l$$ I see this in the application of Davis-kahan theorem to the community detection in chapter 4.5.3 of the book "High dimensional probability" by Roman Vershynin.
My try:
$$ \sin(\angle u,v)\leq l \Rightarrow \sqrt{1-\langle u,v\rangle^2}\leq l$$ $$\|u-\theta v\|^2_2= 2-2\theta\langle u,v\rangle$$
What should I do next?