Here is solution : Enough to solve with finite sequence . Now call the largest number is $ 10N + d$ where $ d\in\{1,..,9\}$ From the condition we must have $ N$ not occur in this sequence . Now if the Number $ 10N + i,\forall i = 0,..,9$ occur then remove it by N. We has : $ \sum_{i = 1}\frac {1}{n_1} + \frac {1}{N} - \sum_{d = 0}^9\frac {1}{10N + d}\geq\sum_{i}\frac {1}{n_i}$ Then repeat this with the sequence ${ \{n_1,...,N}$ then we have result: $ \sum_{i}\frac {1}{n_i}\leq \sum_{k = 1}^9\frac {1}{k}$