Let $\{x_n\}$ be a decreasing sequence of positive real numbers which converges to $0$. Then there exists $m_0\in\mathbb N$ and a constant $S(m_0)>0$ depending on $m_0$ such that $|x_{j+m_0}-x_{j+m_0+1}|\geq\frac{S(m_0)x_{j+1}}{\log^2x_{j+1}}$ for all $j$.
I could not find such a sequence till now. Is such a sequence exists?
Here's an example. Consider $x_n = \frac{1}{2^n}$, then we see that \begin{align} x_{m_0+j}-x_{m_0+j+1} = \frac{1}{2^{m_0+j+1}}= \frac{1}{2^{m_0}}x_{j+1}>\frac{1}{2^{m_0}}\frac{x_{j+1}}{\log^2 x_{j+1}} \end{align} where the last inequality comes from the observation that $\log^2 x_{j+1}\rightarrow \infty$ as $j\rightarrow \infty$.