Let $a_i,b_i,r_i,s_i$ be positive integers for $i\in\{1,2\}$. $r_i$ and $s_i$ are non-zero for $i\in\{1,2\}$.
Let $a=\left(\frac{1}{a_1},\frac{1}{a_2}\right), b=\left(\frac{1}{b_1},\frac{1}{b_2}\right),r=(r_1,r_2),s=(s_1,s_2)$, and $c=\left(\frac{1}{a_1+b_1},\frac{1}{a_2+b_2}\right).$
Let $\langle-,-\rangle $ denote the standard inner product on $\mathbb{R}^2$.
If $\langle a,r\rangle<1$ and $\langle b,s\rangle<1$, does that imply $\langle c,r+s\rangle\leq 1$?