Q:

Verifying Properties of Filters

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! 

A:

I have uploaded a corrected version of the document. 
Thanks again for your close reading. 

A:

There are several typos.  You’re right about the first $\mathcal F$ in the statement of the proposition: it should indeed be $F$. Clause (b) of the proposition should read:

b) $F$ converges to $x\iff\mathcal{F}$ converges to $x$.

The true half of the implication that’s actually written for (b) appears in its proper place as (d).