# Some examples of sequence with cluster points

metric-spaces

Provide the following examples, assuming that $$(X, d)$$ is infinite.

1. A sequence without cluster points.
2. A sequence that has exactly 5 cluster points.
3. 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 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.