reflexive Banach space 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?
 A: We will admit the following result:

A Banach space $X$ is reflexive if and only if for all $l:X\rightarrow \mathbb R$ linear and continuous we can find $x_0$ such that $\lVert x_0\rVert =\lVert l\rVert = \sup_{x\neq 0}\frac{l(x)}{\lVert x\rVert}$. 

Let $l$ such a map. For all $n\in\mathbb{N}^*$, we can find $x_n$ with $\lVert x_n\rVert =1$ and $\langle l,x_n\rangle \geq \lVert l\rVert -\frac 1n$. Let $C_n$ the closed convex hull of the set $\left\{x_k,k\geq n\right\}$. $\left\{C_n\right\}$ is a decreasing sequence of closed convex non empty subsets of $B_X$. Let $x\in \bigcap_{n\in \mathbb{N}^*}C_n$. Let $\varepsilon>0$ and $k_0\in\mathbb N^*$. We can find $N\in\mathbb N$ and $k_1<\cdots<k_N$ integers with $k_1\geq k_0$ and $(\alpha_j)_{1\leq j\leq N}\in \left[0,1\right]$ such that $\lVert x-x'\rVert <\varepsilon$ with $x'=\sum_{j=1}^N\alpha_jx_{k_j}$ and $\sum_{j=1}^N\alpha_j=1$. Since each $k_j$ is greater than $k_0$ we have 
\begin{align*}
\langle l,x\rangle& =\langle l,x-x'\rangle+\langle l,x'\rangle\\
&= \langle l,x-x'\rangle +\sum_{j=1}^N\alpha_j \langle l,x_{k_j}\rangle\\
&\geq -\varepsilon\lVert l\rVert+\sum_{j=1}^N\alpha_j\left(\lVert l\rVert -\frac 1{k_j}\right)\\
&=\lVert l\rVert (1-\varepsilon)-\sum_{j=1}^N\frac{\alpha_j}{k_j}\\
\langle l,x\rangle&\geq  \lVert l\rVert (1-\varepsilon)-\frac 1{k_0}.
\end{align*}
Since the last inequality is true for all $\varepsilon >0$ and $k_0>0$ we get that $\lVert l\rVert\lVert x\rVert \leq \langle l,x\rangle$ and it's an equality (we notice that $x\neq 0$, exect in the case in which $l=0$).
A: Such $C_n$ exist in any Banach space. For example take $C_n$ to be the closed ball of radius $1-1/n$ centred at the origin.
A: It's true This is an old (1939) theorem due to Smulyan Look at any Banach space theory book for the proof 
