I really need your help with the following problem: Let $ N \ge 3 $ be given, then consider $$ L(N)=\max\left\lbrace \sum_{j=2}^{N-1} \frac{c_j}{j} \, \middle| \, c_j \in \mathbb{N}, \nexists 0\le d \le c: 1 \le \sum_{j=2}^{N-1} \frac{d_j}{j}\le 1+\frac{1}{N} \right\rbrace. $$
Prove that $ L(K^2)\le K $ for all $ K\ge 2 $.