A combinatorial problem of Shields and Pearcy

Abstract:

Pearcy and Shields asked the following question. If ${x_1}, \ldots ,{x_n}$ are positive real numbers, can one always delete a subset D (possibly empty) such that the following two conditions are satisfied: (1) $\sum \;1/{x_i} \leqslant n$ (sum over all deleted terms), (2) $\sum \;{x_i} < 1$ (sum over any interval of consecutive terms disjoint from D)? In this note we show that this is always possible.