Let us fix some notations. If $A$ and $B$ are nonempty subsets of a Banach space $X$, we set $d(A,B)=\inf\{\|a-b\|\mid a\in A,b\in B\},\widehat{d}(A,B)=\sup\{d(a,B)\mid a\in A\}.$ Let $A$ be a bounded subset of a Banach space $X$. Set $$\omega(A)=\inf\{\widehat{d}(A,K)\mid \emptyset\neq K\subset X \text{ is weakly compact }\},$$ and $$\omega_{0}(A)=\inf\{\widehat{d}(A,K):\emptyset\neq K\subset A \text{ is weakly compact }\}.$$ I have the following two questions:
Question 1. The two quantities $\omega(\cdot)$ and $\omega_{0}(\cdot)$ are equivalent?
Question 2. There exist a space $X$ and a sequence $(A_{n})_{n}$ of bounded subsets of $X$ such that $\lim_{n\rightarrow \infty}\frac{\omega(A_{n})}{\omega_{0}(A_{n})}=0$?