sqing wrote: Given the sequence $(a_n) $ satisfies $1=a_1< a_2 < a_3< \cdots<a_n $ and there exist real number $m$ such that $$\displaystyle\sum_{i=1}^{n-1} \sqrt[3]{\frac{a_{i+1}-a_i}{(2+a_i)^4}}\leq m $$for any positive integer $ n $ not less than 2 . Find the minimum of $m.$ $$m_{min}=\frac{\sqrt[3] 2}{2} $$

sqing wrote: sqing wrote: Given the sequence $(a_n) $ satisfies $1=a_1< a_2 < a_3< \cdots<a_n $ and there exist real number $m$ such that $$\displaystyle\sum_{i=1}^{n-1} \sqrt[3]{\frac{a_{i+1}-a_i}{(2+a_i)^4}}\leq m $$for any positive integer $ n $ not less than 2 . Find the minimum of $m.$ $$m_{min}=\frac{\sqrt[3] 2}{2} $$ How come?