Let $a \leq b \leq c \leq d \leq e$ be the lengths of the segments. If $a^2 + b^2 > c^2$ or $b^2 + c^2 > d^2$, then we're done. Otherwise, we have \begin{align*}a^2 + b^2 \leq c^2 \quad \text{and} \quad b^2 + c^2 \leq d^2, \tag{$\ast$}\end{align*}implying \begin{align*} c^2 + d^2 &\geq 2a^2 + 3b^2 &&\text{repeatedly using ($\ast$),} \\ &> a^2 + 3b^2 \\ &\geq a^2 + 2ab + b^2 &&\text{since $a \leq b$}, \\ &= (a + b)^2 \\ &> e^2 &&\text{by triangle ineq.}, \end{align*}which finishes it.