Help with Mathematical Analysis: Proof of infinite intersection

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[steady rapids]

first of all, you switched "a" to "x"

also what "<" are you referring to?

you mean the first inclusion?

the part highlighted is typically not a good proof

[hasty junco]

Yeah, correct, should be a

[hasty junco]

Yeah first half of the proof

[hasty junco]

Ow, with tackling these kind of proofs, would you recommend like a standard scheme I can always follow? I'm trying to find somewhat of a pattern I can use for each intersection proof, and/or union proof

Like, a method of proving I always apply for each of these exercises

Cause what troubles me the most is that every proof looks completely different for these proofs, they don't resemble each other at all making it kind of difficult

[steady rapids]

Well typically double inclusion is fine, you are just proceeding in a way I think is strange

Here you have an infinite intersection, and you correctly evaluate said infinite intersection

But you argue in a very strange way as for why the first inclusion holds

For example: how can you go from x < 3 + 1/n (for any n) to x <= 3?

Here, the right way to do it would be to show that you cannot have x > 3 (if that were the case, you would have a contradiction that is fairly easy to find)

[hasty junco]

but how would i know that for this specific exercise this is the right approach

[steady rapids]

for this specific exercise in particular?

well you have an intersection of sets

so any element in said intersection needs to be in all said sets

if there are any sets among those where a value is not there, then you cannot have such a value of x

for example: 3.00000000001

for n large enough, 3 + 1/n < 3.00000000001, so 3.00000000001 is not in ((n+1)/2n, 3+1/n) for said n

so you already know that x cannot be that thing

[hasty junco]

oh but why does this approach not work with this question then

[steady rapids]

well the argument is very rushed

in comparison to the other example you showed

it is true that if u_n tends to L and u_n <= x for all n, then L <= x

but I had assumed you did not see that yet

[hasty junco]

so it's jsut a bad proof over all, isnt it?

[steady rapids]

Let's say it's not the kind of proof that I expect from you at the beginning of a set theory course

[hasty junco]

and one more question im sorry for asking so much haha, does this approach work with any intersections?

[steady rapids]

if you're like 1 year in, then I don't care all that much, because you already know the result L <= x when u_n <= x and u_n -> L

[hasty junco]

ahh okay

yeah this was before we had actual analysis with limits and stuff so kind of weird

but he has never really explained these proofs before which kinda sucks :/

[steady rapids]

well yeah I'm just making you aware of my expectations

if I were to grade

[hasty junco]

would you recommend me to learn a single heart by proof, like the one you just said was pretty good, and try to apply it ot other exercises?

or is that not smart

[steady rapids]

Well I wouldn't say by heart, but apply the same method

[hasty junco]

okay ill try my best thank you for your help man 🙂