What are some examples of sequences that have multiple limit points ? What are some examples of complex or real sequences having more than two accumulation points ?
 A: *

*$\langle 1,2,3,1,2,3,1,2,3,\dots\rangle$ and its congeners. 

*Slightly more generally, let $\{a_0,\dots,a_{n-1}\}$ be a finite set of real numbers, and let $\langle\epsilon_n:n\in\Bbb N\rangle$ be any sequence converging to $0$. Let $x_k=a_{k\bmod n}+\epsilon_k$ for $k\in\Bbb N$; then $\langle x_k:k\in\Bbb N\rangle$ has $\{a_0,\dots,a_{n-1}\}$ as its set of cluster points.

*$\left\langle n\alpha-\lfloor n\alpha\rfloor:n\in\Bbb N\right\rangle$ for any irrational $\alpha$: every real number in $[0,1]$ is a cluster point.

*Any enumeration of $\Bbb Q$ in the form $\langle q_n:n\in\Bbb N\rangle$: every real number is a cluster point.
Added: Let $\langle X,d\rangle$ be a metric space, and let $\langle x_n:n\in\Bbb N\rangle$ be a sequence in $X$. If $\langle X,d\rangle$ converges to $x$, meaning that for each $\epsilon>0$ there is an $m_\epsilon\in\Bbb N$ such that $d(x,x_n)<\epsilon$ for all $n\ge m_\epsilon$, then we say that $x$ is the limit of the sequence. A more informal (but perhaps more understandable) way to say this is that $x$ is the limit of the sequence if every open neighborhood of $x$ contains all but finitely many terms of the sequence.
If, on the other hand, $x$ merely has the property that each open neighborhood of $x$ contains infinitely many terms of the sequence, $x$ is a cluster point or accumulation point of the sequence. Take my first sequence above, for instance: $1$ is a cluster point, because if $V$ is any open interval around $1$, the infinitely many terms $x_0,x_3,x_6,x_9,\dots$ are all in $V$, since they’re all equal to $1$. (My $\Bbb N$ starts at $0$, not $1$.) Cluster points of a sequence are sometimes also called limit points of the sequence, but this is a confusing terminology that ought to be avoided. They can also be called subsequential limits of the sequence, because it can be proved (fairly easily) that $x$ is a cluster point of a sequence $\langle x_n:n\in\Bbb N\rangle$ if and only if some subsequence $\langle x_{n_k}:k\in\Bbb N\rangle$ of $\langle x_n:n\in\Bbb N\rangle$ converges to $x$.
A: Let $z$ be a complex number of norm 1. 

(source: wikimedia.org) 
If $\theta$ is a irrational multiple of $\pi$ the sequence $z^n$ is dense. And if $\theta$ is a rational multiple of $\theta$ the sequence $z^n$ is periodic.
A: $\mathbb{Q_{+}}$ has accumulation points everywhere in $\mathbb{R}$, see link for how to index the positive rational numbers. (Every irrational number is the limit of a sequence of rational numbers).  
