Let $(a_n)_n$ be an increasing real sequence with $a_n=O(\sqrt n)$.
Is it true that there exists an increasing function $\phi:\mathbb N\to\mathbb N$ such that $$\lim \left|\sum\limits_{k=1}^{\phi(n)}\cos(a_k)\right|+\left|\sum\limits_{k=1}^{\phi(n)}\sin(a_k)\right|=\infty?$$
https://artofproblemsolving.com/community/c7h3080371_majoration_of_cosinus_and_sinus_addition
Yes, it suffices that $a_n=o(n)$.
Denote $h(n)=|\sum_{j=1}^n \cos a_j|+|\sum_{j=1}^n \sin a_j|$.
For any fixed integer $M>0$ there exists arbitrarily large $n$ for which $a_{n+M}-a_n<1/M^2$. Thus by triangle inequality and 1-Lipschitz property of functions $\cos$ and $\sin$ we have $$h(n+M)+h(n)\geqslant |\cos a_{n+1}+\ldots+\cos a_{n+M}|+|\sin a_{n+1}+\ldots+\sin a_{n+M}|\\\geqslant M|\cos a_{n+1}|-M/M^2+M|\sin a_{n+1}|-M/M^2\geqslant 2M-2$$ that yields that $h(n+M)$ or $h(n)$ is at least $M-1$. Your claim follows immediately.
