I am trying to prove the above. More specifically I am trying to prove the direction "if all subnets of a given net have limits then the net in question has a limit"
The definition I am using for a subnet is as follows (From Folland)
A subnet of a net $(x_\alpha)_{\alpha\in A}$ is a net $(y_{\beta})_{\beta\in B}$ together with a map $h:B\rightarrow A$ such that
-For every $\alpha_{0}\in A$, there exists $\beta_{0}\in B$ such that $\forall \beta$ such that $\beta_{0}\preccurlyeq \beta$ we have $\alpha_{0}\preccurlyeq h(\beta)$
-$y_{\beta}=x_{h(\alpha)}$
I feel the only way I can approach this is via contradiction. Suppose that $(x_\alpha)_{\alpha\in A}$ did not have a limit, then...then what. I feel I need a way to talk about convergence without knowing what the limit is. In real analysis, one could do this using Cauchy sequences. A quick look on wikipedia shows that a thing called a Cauchy net exists, but since my lecturer gave us this problem without ever talking about Cauchy nets I feel there should be a way to solve this without referring to Cauchy nets.
To be clear, the only things we were taught was basic point set topology and the definition of a net and a subnet, so I would like, if possible, an answer that only utilises these things. A hint or complete answer is welcome.
