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?

## Best Answer

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

everynumber. My tip is to first think of a set of numbers that achieves that andthenabout if it is possible to put them in some order, hence making them into a sequence.