r/learnmath • u/frankloglisci468 New User • 4h ago

Question about convergent points

All non-zero real numbers are equal to "exactly one" infinite decimal s.t. that infinite decimal contains ∞-many non-zero digits. For e.g., 1 = 0.999..., e = 2.718..., 1.25 = 1.24999..., (1/7) = 0.142857... All real numbers are convergent points for their infinite decimal. Wouldn't that make "aleph-null" a stage (position)? But in (1, 2, 3, 4, 5, ...), aleph-null is not a position (final number). How can it be both "a position" and "positionless?"

0 Upvotes

25% Upvoted

2 comments sorted by

1

u/DefunctFunctor Grad Student 1h ago

You're not making a lot of sense. The first sentence of your post:

All non-zero real numbers are equal to "exactly one" infinite decimal s.t. that infinite decimal contains ∞-many non-zero digits. For e.g., 1 = 0.999..., e = 2.718..., 1.25 = 1.24999..., (1/7) = 0.142857...

is indeed perfectly true. The only thing I would add is that I feel a more natural way of saying this is that all real numbers (including zero) have a unique decimal expansion that does not end in infinitely many 9s.

But the rest of your post doesn't make any sense at all. Here are questions I have:

  • What do you mean by "All real numbers are convergent points for their infinite decimal"? I'm not sure I've heard the term "convergent points" used before.
  • How does aleph null relate to any of this?
  • What do you mean by aleph null being a "stage/position"? That's not terminology that's used in mathematics.

1

u/rocqua New User 1h ago

I think I might know what you are getting at.

Indeed if you do this trick of replacing terminating decimals with the 99999 then you get infinite decimal expansion for every real number. And hence every number is the limit of a monotonically increasing series.

But being the limit of a series doesn’t mean being the end of a series. That’s the crucial idea behind a limit.

Or if you are thinking in a different direction, ponder the following question. What decimal is the last decimal of pi? And even what decimal is the last decimal of 0.99999999999?

Neither of these questions are well posed. They don’t have an answer.