[Math] How does a sequence who does not have a limit point behave

Question

In $R$ every bounded sequence have a limit point. Suppose the sequence and bounded and further say that it does not have a limit point. Would this directly conclude that it goes to infinity?

Can't there be sequences who go to infinity but have limit points?

I mean is the statement below true:
A sequence diverges to infinity if and only if it doesn't have any limit points.

Answer 1

Let $(x_n)\subset\mathbb R$ be a sequence of real numbers. Then exactly one of the following two statements is true.

1. $\lim_{n\to\infty}|x_n|=\infty$
2. There exists a subsequence $(x_{n_k})$ of $(x_n)$ that converges to some (finite) real number (i.e. $(x_n)$ has a limit/accumulation point).

If a sequence does not have any finite limit points (so (2) is not satisfied), then this does not mean that $\lim_{n\to\infty}x_n=+\infty$ or $\lim_{n\to\infty} x_n=-\infty$. For instance, $$ x_n=\begin{cases}n, & n \text{ odd} \\ -n, & n \text{ even}\end{cases}=(1,-2,3,-4,5,-6,...). $$ Conversely, if we know that $\lim_{n\to\infty}x_n=+\infty$ or $\lim_{n\to\infty}x_n=-\infty$, then $\lim_{n\to\infty}|x_n|=\infty$, and in this case $(x_n)$ cannot have any finite limit points.