Elementary Measure Theory Question

[ntsivanidis]

Hey guys, below is a small question from introductory measure theory. Maybe be completely wrong on this, so if you could point me in the right direction I'd really appreciate it.

Claim: Let $$B=\mathbb{Q} \cap [0,1]$$ and $$\{I_k\}_{k=1}^n$$ be a finite open cover for $$B$$. Then $$\sum_{k=1}^n m^*(I_k) \geq 1$$

Proof: Let $$\ B = \{q_k\}_{k=1}^\infty$$.Since $${I_k\}_{k=1}^n$$ is a finite cover, there must be at least one $$j \in \{1,\dots,n\}$$ s.t. $$I_j$$ contains infinitely many elements of $$B$$.Fix $$\varepsilon > 0$$. WLOG, WMA $$I_k=(q_k - \frac{\varepsilon}{2(n-1)}, q_k + \frac{\varepsilon}{2(n-1)}) \ni q_k \ \forall \ k \neq j$$. Then $$\sum_{k=1}^n m^*(I_k) = \sum_{k\neq j} m^*(I_k) + m^*(I_j)=\varepsilon + m^*(I_j) \geq m^*(I_j)=1$$ since $$m^*([0,1]\backslash \mathbb{Q})=1 \ \Box$$

 

[Eynstone]

Although the statement is correct, the 'WLOG WMA' part of the proof is too sweepy. You must provide details if the proof is to sound rigorous.

 

[ntsivanidis]

Thanks! Along what lines should I proceed? It's very shaky from that point onwards and I'm stumped for something more concrete. If you could describe a sketch, that would be great. Cheers