Chinglu asserts $$0.999... \neq \sum_{k=1}^{\infty} \frac{9}{10^k}$$ because the latter is "not a thing" but gives no alternate definition.There is no such thing as infinite summation based on natural numbers.

(This was not offered as mathematical proof but as the beginning of a proof. Had someone named such a purported number, it would lead to contradiction. Also, absence of evidence when there must be evidence if the hypothesis is true and a diligent search has been taken is evidence (but not proof) that the hypothesis is wrong.)There is in analysis, because analysis deals with the real numbers, not just the natural, rational, or algebraic numbers.

Here's just one textbook on the subject: http://ramanujan.math.trinity.edu/wtrench/texts/TRENCH_REAL_ANALYSIS.PDF (This was Google's first hit for "analysis textbook" but you are welcome to try any others.)

... [Explicit refutation that one may not have infinite summation, with an applicable example not involving limits] ...

But as long as you are looking at this textbook you might as well look at Theorem 1.1.3 and Theorem 1.1.6 on pages 4 and 6 which says if $$0.999... < 1$$ then there must be a real number equal to 1 - 0.999... and there must likewise be a rational number between 0.999... and 1. The fact that no such numbers have ever been named seems to indicate a problem with the hypothesis $$0.999... < 1$$.

This, by my counting, is post #1532 which may be linked to with [post=3141308]a forum-specific markup like this that does not get your post queued for moderation for too many hyperlinks[/post].I am looking at page 201 of your link and I am glad the mainstream has not contradicted itself on an infinite series.

Chinglu seems to accept my source as authoritative on the subject of infinite summation whereby $$0.999... = 1$$ follows directly. All the rest of this post is chinglu's failure to grasp a subject he's spent less than 5 minutes thinking about in a language he doesn't understand properly.

Also proofsMath is not done by "common sense" nor by popular vote, nor by opinion, BUT BY PROOFS.

*from assumptions and definitions.*You cannot argue with a definition, you can only propose alternatives. The definitions and axioms of the real numbers are fixed and while infinite repeating decimals are rational numbers, you may need a generic theory of infinite decimals to convincingly prove this. Particular for an audience that is dubious that 0.999... / 10 = 0.0999...

This is a stronger claim than chinglu has made above, but is entirely baseless, particularly in light of the above reference. Chinglu doesn't have the right to ignore the mathematical field of analysis just because he is still struggling with set theory.See my response to RPenner.

There is no such thing in mathematics that permits infinite addition.

which thread and number?

As I have said before, you have no right to compel a response from me. You simply misunderstood the text.Sure, #1532 for RPenner. He has not responded.