What the hell?! ???
Well, well … apparently it does!
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 