$\newcommand\ep\varepsilon$The answer is still no (assuming you wanted $c>0$). Indeed, let 
$$s_n:=\sum_{k=0}^n a_k\sin(k\ep),\quad t_n:=\sum_{k=0}^n b_k\sin(k\ep),$$
where $\ep=3\pi/4$, $a_0=b_0=10$, $a_1=1$, $b_1=9$, and $a_k=b_k=2^{-k}$ for $k\ge2$. 

Then $c_1a_k\le b_k\le c_2a_k$ for $c_1=1$, $c_2=9$, and all $k\ge0$. Moreover, $f(n):=s_n>0$ for all $n\ge0$. 

However, $t_1=-9 + 5\sqrt2<-19/20<0$ and, moreover, $t_n<0$ for all $n\ge2$.