# show there exists a strictly increasing sequence satisfying some properties

Consider a strictly increasing (componentwise) sequence $$\{s^k\}_{k \in \mathbb{Z}_+}$$ such that $$s^k = [s^k_1~ s^k_2~ \cdots ~s^k_n]^\mathrm{T} \in \mathbb{R}^n$$, $$s^0 = 0$$ and $$s^k \rightarrow \infty$$ as $$k \rightarrow \infty$$. Does there exist a srictly increasing sequence $$\{r^k\}_{k \in \mathbb{Z}_+}$$ with $$r^k \in \mathbb{R}$$, $$r^0 = 0$$ and $$r^k \rightarrow \infty$$ as $$k \rightarrow \infty$$ such that $$\frac{s_i^{k+1} - s_i^k}{r^{k+1} - r^k} \leq \frac{s_i^{k+2} - s_i^{k+1}}{r^{k+2} - r^{k+1}}$$ for all $$k \in \mathbb{Z}_+$$ and $$i \in \{1~ 2~ \cdots ~ n\}$$?

My strategy is like this: since in this case we have $$r^{k + 2} \leq \frac{r^{k+1} - r^k}{s_i^{k+1} - s_i^k}(s_i^{k+2} - s_i^{k+1}) + r^{k+1}$$ for all $$i \in \{1, \cdots, n\}$$, we can set $$r^1$$ to be sufficiently large, then construct such a srictly increasing sequence $$\{r^k\}_{k \in \mathbb{Z}_+}$$, but the problem is that how can we prove that $$r^k$$ will go to infinity as $$k$$ goes to infinity.