Q:

Sequence with 1 limit point

Suppose the image of some sequence has 1 limit point.
Convergent sequence, for example $a_n = \frac1n$, may have 1 limit point. I wonder, is there any divergent sequence with 1
limit point?

A:

Exactly what do you mean by limit point? If you mean a point that is the limit of the sequence, then of course a divergent sequence of real numbers has no limit point. If you mean a cluster point, however, then the answer is yes: an example is the sequence
$$\langle x_n:n\in\Bbb N\rangle=\langle 0,1,0,3,0,5,0,7,0,9,\dots\rangle$$
in which
$$x_n=\begin{cases}
0,&\text{if }n\text{ is even}\\
n,&\text{if }n\text{ is odd}\;.
\end{cases}$$
This is a divergent sequence with a single cluster point, $0$.

