For a subset $A \subseteq \mathbb{R}$, the extended real number is defined as $$ N(A) = \inf \left\{ \sum_{k=1}^{\infty} \ell(I_k) : A \subseteq \bigcup_{K=1}^{\infty} I_k\right\}$$ where the $I_k$ are open intervals of the form $(a_k,b_k)$. Show that for each set $A \subseteq \mathbb{R}$, it holds that $N(A) = m^*(A)$.
I'm a bit confused, as the definitions are the same. According to me, I should propose a covering and see that the infimum is the same, but since the definitions are identical, it trivially holds.