Sets with one accumulation point

Question

Are there any more examples of sets in $\mathbb R$ that have one accumulation point apart from convergent sequences? I can´t think of any

Answer 1

The set $$A=\Bbb Z\cup\left\{\frac1n:n\in\Bbb Z^+\right\}$$ has only $0$ as an accumulation point, but it is not homeomorphic to a convergent sequence together with its limit point: the latter is compact, and $A$ is not.

Answer 2

Consider the following sequence :

$u_{2n}=n$ and $u_{2n+1}=0$.

It diverges and its only accumulation point is $0$.

Answer 3

Just find a sequence that has multiple sub-sequential limits (1), or has at least one sub-sequential limit and is unbounded (2). For example of case (1):

$$a_n = \begin{cases} \frac{1}{n} \quad \text{ if } n \text{ is even}.\\ 1+ \frac{1}{n} \quad \text{ if } n \text{ is odd}. \end{cases}$$

For an example of case(2), replace $1+ \frac{1}{n}$ with $n$.