Problem 1337 (difficulty: 10/10)

For any \(\displaystyle H\subset\R^p\) bounded set let \(\displaystyle B(H)\) be (a) largest open ball in \(\displaystyle H\) if \(\displaystyle H\) has no interior then let \(\displaystyle B(H)=\emptyset\). Starting from an \(\displaystyle A_0\subset\R^p\) Jordan-measurable set let \(\displaystyle A_1=A\) and \(\displaystyle A_{n+1}=A_n\setminus B(A_n)\). Prove that \(\displaystyle \lim b(A_n)=0\).

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