$0.\overline{9}$ and 1 look different. $0.\overline{9}$ has a zero in the ones place, so it has to be less than 1. $0.\overline{9}$ will always just be nines forever and will never reach 1. $0.\overline{9}$ is way more work to write on paper. I get it.
NO I DON’T FUCKING GET IT. THEY ARE THE SAME. DAMN. NUMBER.
There’s an entire grip of confusion surrounding this topic, so focus up. Going to keep it real simple for you no-math-knowing motherfuckers.
You have $0.\overline{9}$, and you have 1. If 1 is bigger than $0.\overline{9}$, then there has to be a number in between them right? Course I’m right.
Now, think of a number bigger than $0.\overline{9}$ but less than 1. I’ll wait.
…
…
…
If your silly ass is still trying, you can knock that shit off. It ain’t there. No matter what number you think of that’s less than 1, $0.\overline{9}$ is A.L.W.A.Y.S. going to be bigger. Take
$$0.999999999999$$
How about one more 9?
$$0.9999999999999$$
One mo’ ‘gain.
$$0.99999999999999$$
Don’t hurt yourself, slick. We’ll never stop adding nines. So what this this mean? If there’s no number between $0.\overline{9}$ and 1, then they are one and the same. Put them shits on a number line and tell me they aren’t the same goddamn number. Guess what? Not the first time a number’s been expressed 2 different ways, either. That little thing called an equals sign? It says “These two things on either side of me? Same thing.” Observe:
$$\frac{1}{2}=0.5$$
$$2\times2=4$$
$$10^2=100$$
and so on and so forth.
So. There’s no number between $0.\overline{9}$ and 1. That means they are the same. And that’s just peachy, because basically all the fucking math most people ever learn is writing the same numbers in different ways with an equals sign stuck in the middle.
QED* bitches.
*Before you nerds start going all aggro on me, I know this wasn’t a legit proof. Give me some fucking credit. Y’all want to see a true proof with all the
you can handle? Get at me in the comments.
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