Let $ \displaystyle{ \{ (a_n , b_n) : n \in \mathbb N \} }$ a sequence of open intervals on $\mathbb R$ such that $ \displaystyle{[0,15] \subset \bigcup_{n=1}^{n_0} (a_n ,b_n) }$ for some $ n_0 \in \mathbb N$
Prove that $ \displaystyle{ \sum_{n=1}^{n_0} \mu(a_n ,b_n) = \sum_{n=1}^{n_0} (b_n -a_n) >15 } \quad$ where $ \mu$ is the Lebesgue measure on $ \mathbb R$
I need to prove this using induction but I can't see how.