About backwards-defined and forward-defined infinite-length sequences [closed]

Let $$A$$ be a set of sequence starting with $$0$$ followed by an infinite number of terms

Let $$B$$ be a set of sequences whose last term term is $$1$$ preceded by an infinite number of terms

Question 1: Is A intersection B empty?

Question 2: Is it possible/reasonable to define C, directly as a set of sequences that start with $$0$$ and have the last term $$1$$ with an infinite number of terms?

All I know is that A obviously makes sense, and B makes sense if you start defining it from the end and go backwards. I'm not even sure if Question 1 and Question 2 are equivalent.

I don't think your definition of $$B$$ is mathematically corret. If a sequence is preceeded by an infinite number of terms, it doesn't make sense to say that it has a last term. Something that's intuitively equivalent (in some sense) to $$B$$ is "The set of all sequences that converge to $$1$$".

With respect to this re-definition, of course $$A\cap B\neq \phi$$ as the sequence $$0,0+\frac12,0+\frac12+\frac14,\dots$$ is in $$A\cap B$$.

Also, it makes perfect sense to define $$C$$ as the set of all sequences that start with $$0$$ and converge at $$1$$.

Does that help?

• I defined elements of B as $X_0=1$, $X_{-1} =...$, $X_{-2} =...$, $X_{-\infty} =...$. Convergence is equivalent anyway so my question is solved Jun 29, 2021 at 14:38
• @MCCCS Is that a mathematically valid way to define sequences? Can you give me a reliable source that discusses this? If this is a valid way, then I'm sorry, I'm not familiar with this. Jun 29, 2021 at 14:44
• Wikipedia says it is not but I believe what I defined only ignores conventions and not any mathematical laws. Jun 29, 2021 at 14:49
• @MCCCS Don't take this comment negatively, but I think you haven't done any formal analysis courses as of yet which is why you're a little confused about how we deal with infinities. The thing is there are a set of rigorous rules and conventions that we follow in mathematics, and these restrict many things that we thought were perfectly fine when we were in high school. You are quite right that you ignored these conventions. Actually, as far as I know, writing something like $X_{-\infty}$ (which you did) is blasphemy in the mathematical society :). Jun 29, 2021 at 14:56
• Thank you, that's why I felt the need to ask this question Jun 29, 2021 at 16:07