I'm on chapter 5 now in these notes: http://math.uga.edu/~pete/convergence.pdf
I'm stuck trying to prove Proposition 5.6. (on the top of page 21/bottom of page 20).
First, I think that the first $\mathcal F$ that appears in the statement of the proposition should really be a $F$ (where previously $F$ was used for a pre-filter, and then $\mathcal F$ for the associated filter which is generated by taking all supersets of sets which are contained in $F$). So I have been working under this assumption.
My interpretation is that I have two pre-filters on a topological space $X$: $F$ and $F'$, and the two respective associated filters: $\mathcal F$ and $\mathcal F'$.
The statement that I cannot prove is the $(\Leftarrow)$ direction for part (b).
That is, I cannot prove the following statement:
If $\mathcal F'$ converges to $x$, then $F$ converges to $x$.
If anyone has any suggestions on how to prove this I would be very grateful. If necessary I can supply my arguments for the other parts.
Thanks as always!