Let $X$ be a Banach space and $B_X$ be the unit ball.
Suppose that for each $\lbrace C_n\rbrace_{n=1}^\infty\subset B_X$ satisfying $C_n$ are closed convex and $C_n\supset C_{n+1}$ has a nonempty intersection.
Is it true that $X$ is reflexive?
Let $X$ be a Banach space and $B_X$ be the unit ball.
Suppose that for each $\lbrace C_n\rbrace_{n=1}^\infty\subset B_X$ satisfying $C_n$ are closed convex and $C_n\supset C_{n+1}$ has a nonempty intersection.
Is it true that $X$ is reflexive?