Let $H$ be an infinite-dimensional Hilbert space with orthonormal basis $(e_{n})_{n\geq 1}$. Define $x_{N}=N^{-1/2}\sum_{n=1}^{N}e_{n}, \forall N\geq 1$. Let $A$ be the norm closure of the convex hull of $\{x_{N} : N\geq 1\}$.
Question: How do I show $0$ and each $x_{N}$ are extreme points in $A$?
Definition: A point $y\in A$ is an extreme point if for all $y_{1},y_{2} \in A$ and $1<a<1$ such that $y=ay_{1}+(1-a)y_{2}$, we have $y=y_{1}=y_{2}$.
My thoughts: We write $0=ay_{1}+(1-a)y_{2}$ and would like to show that $y_{1}=y_{2}=0$. $y_{1}, y_{2} \in A$ can be written as $y_{1}=\sum_{N=1}^{K}p_{N}x_{N}$ and $y_{2}=\sum_{N=1}^{K}q_{N}x_{N}$ where each $p_{N},q_{N}>0$ and $\sum_{N=1}^{K} p_{N}=1=\sum_{N=1}^{K} q_{N}$ $(K \in \mathbb{N})$.
Thus, $0=ay_{1}+(1-a)y_{2}=a\sum_{N=1}^{K}p_{N}x_{N}+(1-a)\sum_{N=1}^{K}q_{N}x_{N}$. Now, I would think we should substitute with $x_{N}=N^{-1/2}\sum_{n=1}^{N}e_{n}$, but unfortunately, I don't know how to proceed from there.
Follow-up question: How would one go about determining other extreme points in $A$?