let $ n \in \mathbb{N} $ and $ \frac{1}{w_1},\ldots, \frac{1}{w_n} $ for some (not necessarily distinct) $ w_1,\ldots,w_n \in \mathbb{N} $ and $ w_1,\ldots,w_n \ge 2 $ be given. Assume that $ \sum_{i=1}^n \frac{1}{w_i} = W + r $ with $ W \in \mathbb{N}, W \ge 5 $ and $ \frac{1}{8} \le r < 1 $. Show, that it is possible to find $ K \subseteq \lbrace 1,\ldots,n \rbrace $ such that $ \sum_{j \in K} \frac{1}{w_j} \ge 1 $ and $ \sum_{j \notin K} \frac{1}{w_j} \ge W-1 $.
Can the lower bound $ \frac{1}{8} $ for $ r $ be improved such that the statement is still true?