or,

      x = 0.9999...
    10x = 9.999...
    10x = 9 + 0.999...
    10x = 9 + x
     9x = 9
      x = 1

Presented slightly more clearly

https://en.wikipedia.org/wiki/0.999...#Algebraic_arguments

Yeah that’s way more complicated than it needs to be and I’m tempted to replace that whole section with:

x = 0.9999...

2x = 1.9999...

2x - x = 1

x = 1

It's much more intuitive that 100.9999...=9.9999... than that 20.9999...=1.9999...

You've met the formatting limitations. You mean:

"It's much more intuitive that 10 * 0.9999... = 9.9999... than that 2 * 0.9999...= 1.9999..."

How do you jump from

x = 0.999...

to

2x = 1.999..

Picture an infinite ticker tape in your mind.

Write

   0.9
 + 0.9
 —————-
   1.8

Now keep extending the 9’s. You have to carry a one, so fix the carry and then add more nines.

Where people keep getting tripped up is thinking you can stop when you get tired, or die, or when the universe ends in heat death. You don’t get to stop. You never get to stop. It’s nines all the way down.

It’s less intuitive, but still relatively straightforward from performing the “long” multiplication. You do need to convince people that the “8 at the end“ is irrelevant, however.

You need to convince people there is no 8 at the end because there is no end.

If you see an 8 it’s because you didn’t carry the 1. Keep going.

Another way to think about it is that you have n digits, and I’m using the n and n + 1 digit at the same time. But since n goes to infinity, +1 hardly matters.

The difficult part is to convince that (9.9999.... - 0.9999....) = 9. Someone else mentioned about Zeno's paradox involving Achilles and the tortoise. Since you multiplied x by 10 to get 9.9999...., you know that it's always going to be 1 decimal place slower in pace to reach convergence compared to 0.9999.... What you might get instead is (9.9999.... - 0.9999....) = 8.9999....

I like this one better:

    0.9        = 1 - 0.1
    0.99       = 1 - 0.01
    0.999      = 1 - 0.001
    0.9999     = 1 - 0.0001
    0.99999... = 1 - 0.00000... with a 1 at the end of the infinite series of 0

>[...] at the end of the infinite series of [...]

...uhh

"Eternity feels a bit too long ... especially near the end."

> 10x -x

subtracting infinities is dangerous, you can achieve any result from it

https://www.youtube.com/watch?v=-EtHF5ND3_s

You're not subtracting anything infinite. Whatever you think of 0.99999... (and the correct thing to think is that it equals 1), I hope we can both agree that it is at least finite! I mean, we can agree that it's less than 2 and more than 0, right?

That subtraction is just as valid as saying 0.333... + 0.333... = 0.666..., or that 1/3 + 1/3 = 2/3.