Some examples of sequence with cluster points Provide the following examples, assuming that $(X, d)$ is infinite.

*

*A sequence without cluster points.

*A sequence that has exactly 5 cluster points.

*A sequence $(x_n)_n$ such that every $x \in X$ is a cluster point of $(x_n)_n$.

I have to do these exercises for my math class.
I thought that in step 1: $x_n = n$ has no cluster point.
For the remaining two points I'm stuck. Can you help me by providing some examples with an explanation?
 A: A cluster point can be thought of as the limit of a subsequence. Here is a trick to make sequence with exactly two cluster points (in the space $X = \mathbb{R}$). It will help you with question 2. Question 3 is trickier.
We start with a sequence with a limit. For instance: $1, 1/2, 1/3, 1/4, \ldots $ which has limit 0.
Then we take a second series that also has a limit, but a different limit. For instance the sequence $1, 3/2, 7/4, 15/8, \ldots$ which has limit 2.
Now to make your sequence $x_n$ we just 'interweave' them:
$$1, 1, 1/2, 3/2, 1/3, 7/4, 1/4, 15/8, 1/5, 31/16, \ldots$$
This has two cluster points: 0 and 2 because for each clusterpoints there is an infinite subsequence that has it as its limit.
What the example also illustrates is that sequences need not be monotonic: they can jump up and down according to a pattern that need not be clear at first glance.
For question 3 you want to get arbitrarily close to every number. My tip is to first think of a set of numbers that achieves that and then about if it is possible to put them in some order, hence making them into a sequence.
A: A cluster point of a sequence $(x_n)_n$ is the limit of some subsequence $(x_{n_k})_{n_k}$. Hence the simplest example I can think for 2. are 5 different constant sequences $(x^i)_n$ in $(X, d)$ put together like this: $x^1,\ldots,x^5, x^1,\ldots,x^5,\ldots$. Of course, the sequences don't need to be constant, you can use, for example, $x^1_n := a^1 + 1/n,\ldots, x^5_n:= a^5 +1/n$ for fixed $a^i$'s. The key idea is to intercalate different convergent sequences.
For 3., think of the metric space $\mathbb{R}$ with the euclidean distance. Since $\mathbb{Q}$ is dense in $\mathbb{R}$, if you could find a way to enumerate all the rationals you will have a properly defined sequence where any $x\in\mathbb{R}$ is the limit of some subsequence. The problem now reduces to finding an explicit bijection from $\mathbb{N}$ to $\mathbb{Q}$.
