Is it possible to use Axiom of Choice to prove that there would exist a sequence (A_n)_{n=1,2,3,\ldots} with the properties: A_n\subset\mathbb{R} for all n=1,2,3,\ldots,
A_1\subset A_2\subset A_3\subset\cdots
and
\lim_{k\to\infty} \lambda^*(A_k) < \lambda^*\Big(\bigcup_{k=1}^{\infty}...