I have proved that if $I_1,\cdots,I_k$ are open sub-intervals of $[a,b]$, then $|I_1 \cup \cdots \cup I_k| \leq |I_1|+...+|I_k|$, where |I|=length of the interval I. Assuming $m[a,b]=0$. Given $\epsilon>0$, then $|I_1|+\cdots+|I_k| < \epsilon $ and $|[a,b]|=|I_1 \cup \cdots \cup I_k|+|a|+|b|=|I_1 \cup \cdots \cup I_k|=b-a$, arriving at a contradiction.
Is this correct?