Akira

I feel that my question is wrongly marked as duplicate. Could you @amWhy please have a re-check on it?

Honestly, such act on my question really hurts my heart.

My question was answered almost 1 month ago. Honestly, I feel this question is clear. I could not understand why it is closed recently. At first, it received 3 upvotes. In recent days, it has received 3 downvotes.

3 , 4 , 5. Could you please have a look at them and give me some suggestions?

Recently, I have asked some questions about continuous-time Markov chain and received no attention. 1, 2 ,

Hi everyone! I have asked a question about a closed formula of Cantor set here. It seems to me that nobody is interested in this question. Could someone help me resolve the question? Thank you so much!

Recently, I have asked some questions on continuous Markov chain, but none of them receives attention. math.stackexchange.com/questions/3648887/…, math.stackexchange.com/questions/3644829/…, math.stackexchange.com/questions/3640094/…. Is there any problem with how I ask the questions?

It's very ridiculous to me that we have a tag that can never be appeared.

How can I get just proof-verification?

@lonestudent, I picked just proof-verification.

Why there are two tags although they will be changed to just solution-verification?

Hi everyone! I have asked a question about a closed formula of Cantor set here. It seems to me that nobody is interested in this question. Could someone help me resolve the question? Thank you so much!

Hi everyone! I have asked a question about a closed formula of Cantor set here. It seems to me that nobody is interested in this question. Could someone help me resolve the question? Thank you so much!

Best wish for you!

I have finished typing my proof at math.stackexchange.com/a/2987245/368425. If you don't mind, please have a check on it. Thank you so much!

Hi

In the same direction you suggested

OMG I'm able to prove it

:)

Yeap, I got that. I'm trying to prove that

But I'm still interested in the author's proof

Although I have successfully proved the theorem, I do it by another approach, which is different than that of the author.

Thank you so much!

Honestly, I have not been able to prove the converse direction. I quite don't understand what you meant by But the converse direction is easier, as you yourself noted. Could you explain how to prove that all ordinals less than $α+\sup\limits_{β∈A}(β)$ is also less than $\sup\limits_{β∈A}(α+β)$?

Thank to your elaboration, I got what that sentence's meaning. But I think the author's proof is not sufficient and not complete. Although he has proved that all ordinals less than $\sup\limits_{β∈A}(α+β)$ is also less than $α+\sup\limits_{β∈A}(β)$, he has done NOTHING in proving the converse direction that all ordinals less than $α+\sup\limits_{β∈A}(β)$ is also less than $\sup\limits_{β∈A}(α+β)$. As a result, he can not claim that Therefore the ordinals less than $\sup\limits_{β∈A}(α+β)$ and the ordinals less than $α+\sup\limits_{β∈A}(β)$ are the same. Thus he has not completed the proof.

Thank you so much for your answer! It seems to me that you meant $\alpha + \sigma = \alpha + \underset{\eta < \sigma}{\lim}\eta$ rather than $\underset{\eta < \sigma}{\lim}\alpha+\eta$, and $ \alpha + \underset{\eta \in A}{\lim}\eta$ rather than $\underset{\eta \in A}{\lim} \alpha + \eta$. On the basis of your answer, I have presented a detailed proof and posted it as an answer below. If you don't mind, please have a look at it. Thank you so much!

Don't bother you any more, feel free to do your work @Holo

When I finish my textbook, i will come back to your problem and give it a try

too much for me

Great! I have seen a proof that requires finite character before and I stay back ^^

It seems to equivalent to AC

I heard that theorem some times on MSE

Oh I see, we build $\Bbb R$ by axioms, not constructively

Your questions is out of my reach in the meantime :)

It's hard for me to understand that. I meant how do we actually understand the theorem that every bounded sequence has a limit without knowing the completeness of $\Bbb R$

It seems to me that we can not understand Analysis without knowing how to build number systems

You know actually my I just study Set Theory as a stepping stone to study Analysis and Topology. I hope that I will soon finish it to get to the main courses

I have a close friend who is a manga fan but he does not know that movie

But you did know that movie, that's interesting

^^ You are in the opposite polar to me in that respect

I love that name

Bravo Yes it is

Do you know character Akira?

Good catch! You have seen that comment before I realized that mistake and deleted it

I'm very curious about how you know Akira is acutually me

Thank your for your sympathy :)

It's now a tough period of time and i hope that i will soon get over it. After that all things will become as usual.

I know it's not fair to do so

6 questions per 24h, and 50 questions per 30 days

Oh ^^ Since you don't need to ask questions like me ^^