What the hell?! ???

Yes, it’s true. The proof shown in the OP is the standard one. Another proof can be framed in terms of the sum of a convergent geometric series with the ratio of consecutive terms equal to ^{1}/_{10} = 0.1 (exact). (As an aside, this counterintuitive fact illustrates that infinity in mathematics is not as simple a concept as one might think.)

A mathematically less rigorous way to think about it is as follows:

^{1}/_{3} = 0.333333…

Then ^{1}/_{3} + ^{1}/_{3} + ^{1}/_{3} = 0.333333… + 0.333333… + 0.333333… = 0.999999…

But ^{1}/_{3} + ^{1}/_{3} + ^{1}/_{3} = ^{3}/_{3} = 1.

Therefore 0.999999… = 1. QED.

'Luthon64

Very convincing for sure. It’s somehow more intuitive than the proof in the OP.

Thanks, Mefiante. 8)

I saw this first with dawkin’s scale discussions

So I follow the first proof untill here:

a = 0.999…

10a = 9.999…

10a = 9 + 0.999…

10a = 9 + a

why is 9a now 9 please?

From your last step above:

10a = 9 + a

=> 10a - a = 9 + a - a (subtract a from both sides)

=> 9a = 9

Thank you, briandvds, I follow that