Showing that the following definitions of Limit Superior are Equivalent

Question

In our Real Analysis textbook by Robert G. Bartle. I came across two different definitions of the Limit Superior ($LimSup$).

1. If $x$ be the Limit Superior of the sequence $x_n$ then for every $\epsilon>0$ $$x_n<x+\epsilon$$ ultimately. And $$x_n>x-\epsilon$$ frequently.

2. If $S$ be the set of all Subsequential Limits of $x_n$, then the Limit Superior of $x_n$ is defined as : $$Sup(S)$$

I cannot understand how to rigorously prove that the statement $1$ can imply statement $2$. Any ideas about how to approach the proof would be really helpful. Thanks in advance.