I couldn't figure out where else to post this so here it is. I'm working through my lecture notes for my analytical statistics course, currently covering the topics of convergence in probability and distribution, and then this comes up. Spoiler: Picture Is there even a possibility that this proof is valid given that the second line uses a result as a condition??? I've done some googling and I've found nothing so far that even comes close to following this process. Normally I would just "oh that's a bad proof, whoever wrote it is wrong". But this is my statistics lecturer who has a doctorate in the subject and for a lecture for an accredited university so like? There has to be something that I'm missing, right? Edit: changed professor to lecturer because I continue to forget that they are not the same person.

I'd email your professor and ask about the fact that the second line uses a result as a condition. But also professors can still be very wrong.

I've been thinking about doing that but 1. the lecturer and the professor are not the same person (I have 0 idea why) and 2. the professor, the person to whom I must direct all correspondence and the person who did not make the powerpoint, has been...Bad about giving informative answers.

Fun times! I'd send the email anyway. Even some kind of response seems like it'd be informative in this case? Or show up at office hours. Or honestly compare what you can find on the web as general stuff on convergence in probability. I would try to help with the actual stats but my eyes glaze over. SPSS gives pretty outputs so I don't have to brain and that's the way I like it.

It's a statement that logically has to be true. The statement is, since Xn is either less than, equal to, or greater than x, saying "Xn <= x-epsilon" is equivalent to saying "either (Xn <= x-epsilon and Xn<=x) or (Xn <= x-epsilon and Xn>x)." It's not invalid, just unsurprising.

Looking at the truth tables, it does not seem to be a tautology unless I ran them wrong. Spoiler: Picture

Yeah, I'm finding nothing online that's using this same logic. And yeah I feel that very hard. Reading math takes SO much focus from me, it's kind of wild.

The statement made in the proof on line 2 is P <- -> ((P^Q) v (P^~Q)) Add that column to your truth table and I think you'll find it's true in every case.

NO WAIT, it's just a logical equivalence of P. So that means of course there's a two way implication.

Yeah, it's one of the (many) reasons I hate this lecturer so much. None of the others do their things like this. Anyway, thank you, I would definitely not have figured this out, because for whatever reason I had not been introduced to tautology before.

Have you taken propositional logic? It was in the philosophy department in my school but it's really useful if you're getting seriously into proofs. And it might count toward distribution requirements despite being essentially a pure math course.

No, all of my logic education came from set theory in undergrad, which was apparently decided by the school to be intro to logic, intro to mathematical proofs, intro to set theory, and just a dusting of intro to combinatorics for flavor all in one. So we learned our truth tables and our proof techniques for algebra and calculus, and all of that in the first quarter of the semester. Then from there we went on to just focus primarily on set theory itself.