How is the limit infimum of sets different from the limit infimum of a sequence of real numbers? [duplicate]

I am trying to understand whether the limit infimum of a set is related or a generalization of the limit infimum of a sequence of real numbers. Suppose that $X_n$ is a sequence of sets, and so the limit infimum of $X_n$ is defined as saying that from some stage onwards, all of the $X_n$ occur.

Concretely:

$$\liminf_{n \to \infty}X_n = \bigcup_{n=1}^{\infty}\bigcap_{k\geq n}X_k$$

Now, suppose that we let $X_n(\omega) = (-1)^n$ and let $X(\omega) = 1$. Then, we have that:

$$\liminf_{n \to \infty}X_n = -1$$

Here it appears that after some time, only $-1$ occurs. But, I know that both $1$ and $-1$ alternate, and so there shouldn't be a time after which $-1$ only occurs.

I am wondering if I am somehow confusing something here between the limit of sets and the limit of a sequence? It also seems that the definition of the limit infimum/supremum of sets has no direct connection to that of a sequence and is an arbitrary definition. Am I wrong here?

marked as duplicate by Did limits StackExchange.ready(function() { if (StackExchange.options.isMobile) return; $('.dupe-hammer-message-hover:not(.hover-bound)').each(function() { var$hover = $(this).addClass('hover-bound'),$msg = $hover.siblings('.dupe-hammer-message');$hover.hover( function() { $hover.showInfoMessage('', { messageElement:$msg.clone().show(), transient: false, position: { my: 'bottom left', at: 'top center', offsetTop: -7 }, dismissable: false, relativeToBody: true }); }, function() { StackExchange.helpers.removeMessages(); } ); }); }); Oct 29 '16 at 12:16

• Sorry, what's $\omega$? – yo' Oct 29 '16 at 11:58