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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!

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    While including the link to the notes, *please* do your best to make questions self-contained. Include the content of the said proposition.2012-05-23
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    Please forgive me if I come across negatively here, I only am saying this because this issue has been brought up regarding my questions in the past. In this case I think the question is self-contained (aside from including definitions), despite the fact that it refers to the source from which the question came. I had however only extracted the part of the problem that I was having trouble with.2012-05-23
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    No, a self-contained question would have a part quoting the proposition at hand. It is not always a one-click thing to open a .pdf file.2012-05-23
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    I'm not trying to argue, but I still believe my question statement meets the criteria you supply. "The statement that I cannot prove is the $(\Leftarrow)$ direction for part (b). That is, I cannot prove the following statement: If $F′$ converges to $x$, then $F$ converges to $x$."2012-05-23
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    You have to remember that you have *more* details than me. What you may seem to think is self-contained may not actually be self-contained. Trust me on that, this is why I never got a 100% in an exam yet (nor in most of my works so far) and I am a bit of an expert in the topic of "hidden information". The problem with this question is that a reader refusing to open the .pdf link has to work **very** hard to fully understand the problem. Adding another few lines of quote would make the post completely readable at the first go - which is something you should *always* strive for as a writer.2012-05-23
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    I agree with you 100%. I should certainly make every effort to ensure that the *person who is in no way obligated to help me but nonetheless is donating their time for my benefit* has no unnecessary difficulty in understanding my question.2012-05-23
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    I must confess that I did not care to open the PDF. Then stopped reading the question. You could simply state what you want to prove instead of saying on which page it can be found.2012-05-24
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    I included the link because it seemed as if there may have been one or more typos in the original proposition. The proposition consisted of 5 claims, only one of which I was asking about. It is unfortunate people dismiss these minor omissions as laziness on the part of the poster. There are repetitive stress (tendonitis/CTS) injuries out there that many people are dealing with. And typing 'tex' code in particular seriously aggravates these conditions. This is my biggest motivating factor in selecting what material is important to *duplicate*.2012-05-24
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    That being said, it is worth noting the part I *was* asking about was reproduced in full. I still feel that my question was completely self-contained. Anyways, I do not wish to be argumentative (though I probably already have been here). Please trust that I will think extensively about this when posting any further questions.2012-05-24

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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).

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I have uploaded a corrected version of the document.

Thanks again for your close reading.

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    You might want to take a look at the proof of Theorem 3.6; the gap noticed [here](http://math.stackexchange.com/questions/144099/proof-of-kelleys-theorem) is still there.2012-05-25