Extreme points of the unit ball of the space $c_0 = \{ \{x_n\}_{n=1}^\infty \in \ell^\infty : \lim_{n\to\infty} x_n = 0\}$ I want to prove that all "closed unit ball" of
$$
c_0 = \{ \{x_n\}_{n=1}^\infty \in \ell^\infty : \lim_{n\to\infty} x_n = 0\}
$$
do not have any extreme point. Would you please help me?
(Extreme Point) Let $X$ be a vector space and $A \subset X$ be convex. We say $x\in A$ is an extreme point if for $x = (1-t)y + tz,\; y,z,\in A, \;t\in(0,1)$ then $y = z = x$.
What I tried is as follows:
Let $B$ be a closed unit ball of $c_0$, that is, $$B = \{\{x_n\}_{n=1}^\infty \in \ell^\infty : \lim_{n\to \infty} x_n = 0 \text{ and } \|x\|_{\ell^\infty}\le 1\}.$$
If there is a extreme point $b = \{b_n\}_{n=1}^\infty\in B$, then we have for
$$
b = (1-t)y + tz, \quad y,z\in B,\quad t\in (0,1)
$$
implies
$$
y = z = b.
$$
But I cannot do anymore here. Would you please help me?

Hint: if $x$ is in the unit ball of $c_0$, there is some $i$ such that $|x_i| < 1$.  What happens if you increase or decrease $x_i$ a little bit?
