A net in $\mathbb{R}$ Let $\{x_j\}_{j\in J}\subset \mathbb{R}$ be a net, $J$ is a directed set. 
If $\{x_j\}_{j\in J}$ does not converge to 0, then there is a subnet$\{x_b\}_{b\in B}$, $B$ is a directed set, that $x_b\rightarrow x,$ where $x$ is either $\infty,-\infty,$ or a nonzero real number.
I see this in a proof of the theorem below:

Any finite dimensional topological vector space has the usual Euclidean topology.

I can only prove that there is a convergent subnet, but not surely has nonzero limit, by using the statement below:

A net $\{x_j\}_{j\in J}$ has a cluster point y if and only if it has a subnet $\{y_\beta\}$ that converges to y.

How to prove there is a convergent subnet with nonzero limit?
Any help would be appreciated.

Classical fact: if every subnet (in any space $X$) of a net has itself a subnet that converges to some fixed $p$, then the original net converges to $p$. (This can be proved using the cluster point fact, if you like.)
Suppose we have net $(x_j)_{j \in J}$ that does not converge to $0$. As $[-\infty, +\infty]$ (the two point compactification of the reals) is compact, there is some subnet that converges to some $p \in [-\infty, +\infty]$ (and hence so do all of its subnets). By the first classical fact, we can in fact choose such a subnet for some $p \neq 0$, or else the net would have converged to $0$ in the first place. QED.
