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.


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