1=0.9999...

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