does .999999999999.... = 1?

My asshole friend keeps trying to tell me that .9999999... = 1

So, what's your take?

Yes, there are several more or less elegant explanations for this. One I find particularly intriguing is based on the notion that on |R (real Numbers), you can always find a number between two numbers, i.e. there exists an y with x < y < z for all x, z with x not equal z. Therefore, if we cannot find an y, x equals z.

Let's just try this out and see what happens:
i) If we try to find a number between 0.999 and 1, our first idea would be to just append another 9 to 0.999, giving 0.9999, which leads us to: 0.999 < 0.9999 < 1, which is correct.
ii) We can do this ad nauseam: Appending one more 9 still fits x < y < z
iii) But 0.999... has infinitely many 9's -- we just can't append more 9's to create a number between it and 1, because 9 is the largest digit we can use in base 10, leading us to the answer to the question whether there is a y for:

0.999... < y < 1; 0.999 != 1

The answer is NO.
As we proposed earlier, this directly leads us to
0.999... = 1



There are countless other proofs, but I find this one especially intriguing, given its non-reliance on actually calculating anything.