Take a net $s: D \rightarrow (X,T)$ and then its associated filter, $\mathcal{F}_s$. Then we take the associated net of this filter, $s_{\mathcal{F}_s}$. Do $s$ and $s_{\mathcal{F}_s}$ coincide?
It makes sense that they should, otherwise our definitions of filters and nets leave something to be desired, however, $s_{\mathcal{F}_s}$is dependant on the containment of sets in $\mathcal{D_\mathcal{F}}$ which isn't much to do with the original ordering.
Is the statement true and how can you prove it?
Thanks :)
It's true, in essence. The nets cannot be equal literally, because $s$ has $D$ as its indexing set (with some directed preorder) while $s_{\mathcal{F}_s}$ has an entirely different indexing set (usually some product of the filter and the set of points), not $D$. And a net is essentially a function and so these functions with different domains are not equal.
What is true is that they are subnets (specifically so-called AA-subnets) of each other. This is sometimes called being equivalent nets. A cluster point (or limit) of one is a a cluster point (limit) of the other and vice versa. So for "practical", "theorem proving" purposes they can often be treated as the same thing. See the book by Schechter on the foundations of analysis for more on such matters.