Let's call that lenghts $a_1,...,a_5$ and $a_1+...+a_5=s$. Then, inequality can be written as $\sum_{i=1}^{5}\frac{a_i}{s-a_i} <2$ For $i=1,...,5$, from triangle inequality follows $a_i < s - a_i$, or equivalently: $s-a_i > \frac{s}{2}$. So applying this inequalitiy on each denominator we have: $\sum_{i=1}^{5}\frac{a_i}{s-a_i} < \sum_{i=1}^{5}\frac{2a_i}{s}= 2$