Cluster point for a net Let $X$ be a topological space. Let $\langle x_{\alpha}\rangle_{\alpha \in A}$ be a net in $X$. For $\alpha$, define $E_{\alpha}=\left\{x_{\beta}:\alpha\lesssim \beta\right\}$.
Is it true that :
$x$ is a cluster point of $\langle x_{\alpha}\rangle_{\alpha \in A}$ if and only if  $x\in \cap_{\alpha\in A} \overline{E_{\alpha}}$.
Here $x$ is a Cluster point for the  net $\langle x_{\alpha}\rangle_{\alpha \in A}$ if for every neighborhood $U$ of $x$ ,  $<x_{\alpha}>$ is frequently in $U$.
Thanks in advance.
 A: Ok, so we want to show that $x$ is a cluster point if and only if $x \in \cap_\alpha \overline E_\alpha$.
Definition : $x$ is a cluster point of $x_\alpha$ if for every neighbourhood $U$ of $x$, $x_\alpha$ is frequently in $U$.
Translating this mathematically gives : $x$ is a cluster point of $x_\alpha$ if for every neighbourhood $U$ of $x$ and $\alpha_0 \in A$, there is some $\beta \geq \alpha_0$ such that $x_\beta$ is in $U$.

Let us start with $x$ as a cluster point. We want to show that it is in the intersection of $\overline E_\alpha$, so fix some $\alpha_0$. We want to show that $x \in \overline E_{\alpha_0}$, so we will pick any neighburhood of $x$, call it $U$. By definition of cluster point, there is some $\beta \geq \alpha_0$ such that $x_{\beta} \in U$. But then, $x_{\beta} \in E_{\alpha_0}$.
So what have we shown? We have shown that every neighbourhood $U$ of $x$ has non-empty intersection with $E_{\alpha_0}$. Thus, $x$ is in the closure of $E_{\alpha_0}$, and now $\alpha_0$ was fixed but arbitrary, so this statement is true for all $\alpha \in A$, showing one direction.

The other way, suppose that $x \in \cap_\alpha \overline E_{\alpha}$. We want to show that $x$ is a cluster point. So fix any neighbourhood $U$ of $x$, and any $\alpha_0 \in A$. We know that $x \in \overline E_{\alpha_0}$, so that any neighbourhood of $x$, in particular $U$, has non-empty intersection with $E_{\alpha_0}$. So there is some $x_{\beta} \in E_{\alpha_0}$ which belongs to $U \cap E_{\alpha}$, but then $\beta \geq \alpha_0$ by definition of $E_{\alpha_0}$. 
What have we shown? We have shown that for every neighbourhood $U$ of $x$ and $\alpha_0 \in A$, there is $\beta \geq \alpha$ such that $x_{\beta} \in U$. This is the definition of cluster point.
Hence, we complete the proof. 
