This is a question from my analysis book. It asks to prove for open sets $G_k$ and Lebesgue measure $\lambda$: $$\lambda \left( \bigcup_{k=1}^\infty G_k\right) \leq \sum_{k=1}^\infty \lambda(G_k)$$ It asks to proceed by noting that $(a,b) \subset \cup_{k=1}^\infty G_k$ implies $b-a \leq \sum_{k=1}^\infty \lambda(G_k)$ by considering the fact that $$[a+\epsilon, b-\epsilon] \subset \bigcup_{k=1}^N G_k$$ for sufficiently small $\epsilon$ and large $N$.
I see how to prove the above fact but I cannot get the sub addivity inequality just by satisfying it for an interval contained in the union of the open sets $G_k$. How to proceed? Any pointers?