Cluster points elaboration I am having difficulty understanding the concept of cluster points.
 Some examples I found that I need help in understanding and why it is the case are:

  
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*A point $x\in \mathbb{R}^p$ is a cluster point of $A$ if and only if for every natural number $n$ there exists an element $a_n\in A$
  such that $0<||x-a_n||<1/n$.
  
  
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*A finite subset of $\mathbb{R}^p$ has no cluster points.
  
  
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*Let $B=\mathbb{I}\cap \mathbb{Q}$ be the set of all rational numbers in the unit interval. Every point of $\mathbb{I}$ is a cluster
  point of $B \in \mathbb{R}$, but there are no interior points of $B$.
  
  


By definition, a cluster point of a set $S\subseteq X$ is a point $x\in X$ such that for every neighbourhood $U$ of $x$, there is a poit $y\in S\cap U$ with $y\ne x$.
For (1) note that taking $U=\{y\in \mathbb R^p\mid \Vert x-y\Vert<\frac1n\}$ shows the existence of such $a_n$. On the other hand, if such a sequence exists and $U$ is any open neighbourhood of $x$, then it contains an open ball of some positive radius $\epsilon$ around $x$; for $n>\frac1\epsilon$ we find that $a_n\in U$, thus showing that $x$ is a cluster point.
For $(2)$, if $A$ is finite and $x\in\mathbb R^p$ arbitrary, let $r=\min\{\Vert x-a\Vert \mid a\in A\setminus\{x\}\}$. Then $r>0$ because we take the minimum of finitely many positive numbers and hence the $r$-ball around $x$ contains no elements of $A$ (apart from posibly $x$ itself), thus showing that $x$ is not a cluster point.
For (3) let $x\in \mathbb I$ and $\epsilon>0$ (wlog. $\epsilon<1$). If $n>\frac1\epsilon$, then let $k$ be the least integer $>nx$ and $k'$ the greatest integer $<nx$. Then $n(x-\epsilon)<k'<nx<k<n(x+\epsilon)$, hence $\frac {k'}n,\frac kn\ne x$ is in the $\epsilon$ neighbourhood of $x$. Since $k>nx\ge 0$, $\frac kn\ge0$. Show that at leats one of these is in $\mathbb I$ (i.e. we cannot have $\frac {k'}n<0$ and $\frac kn>1$ at the same time).
There is no interior point for a similar reason: every open interval contains an irrational number (e.g. a number of the form $y+\sqrt 2$ with $y\in\mathbb Q$).
