1 = 0.999..., the Conclusive Argument
Many early math students find the equality 1 = 0.999... confusing, surprising, some down-right think it must be obviously wrong.
Here I will give the conclusive, in-arguable proof that this equation is in fact true. A lot of other people have done the same, and there are many valid and well explained proofs of this equality.
But what I'm going to do, that is different, is to argue this by almost no argument at all!
Because in spite of the abundance of clear and valid proofs, with intuitions well supplied, and so on -- people still try their darnedest to argue against the equality. And in argument after argument that I have seen, they all seem to come down to a very basic misunderstanding.
As far as I can tell, the only reason people misunderstand this equality, is because they misunderstand what is meant by the expression 0.999...
(Ok, some people misunderstand the equality because they have some strange religious beliefs, or some kind of mental disorder that makes them interpret this equation in weird ways. There are some strange people out there. But for mentally normal people, their issue with this equality seems to be just about misunderstanding the meaning of 0.999...)
What Does 0.999... Mean?
Let's start with what it doesn't mean. It does NOT mean "almost 1" or "very close to 1". It kind of means "infinitely close to 1" but what does that mean exactly? Let's be a little more precise with our language than that.
Technically speaking, 0.999... is just notation for a limit. If you want to go deep on what a limit is, probably the best thing to do is watch a YouTube video like this one, or pick up a book on Calculus or Real Analysis.
But I'm going to keep the description as simple as possible. When talking about a limit you always have to start from a sequence. Let's take for example the sequence 0.9, then 0.09, then 0.009, and so on.
The limit of this sequence is a number, and the sequence gets close to it, eventually. I know "close to" doesn't sound like a rigorously defined mathematical phrase, so let's make this more rigorous. When I say that the sequence "0.9, 0.99, 0.999, and so on" gets close to 1, I mean this collection of facts:
Eventually the distance from 1 is less than 0.1, and eventually the distance from 1 is less than 0.01, and eventually the distance from 1 is less than 0.001, and so on.
Limits
So to be as clear as possible: The limit of a sequence is a number, such that the sequence gets close to the number.
And getting close to the number means: Pick any distance you like, and eventually the sequence is within that distance.
The Proof
So finally, here is the proof that 1 = 0.999... without hardly any proof:
This equation just means that the sequence 0.9, then 0.99, then 0.999, and so on, gets close to 1. And that's just obviously true!
Ok But
Yes, that is not a rigorous proof, but my intention here wasn't to give a rigorous proof. As I said, that already exists elsewhere. And I could give it here too.
But my point is that most people don't really need a proof when they're trying to understand why this equality is true. What most people need is actually just an explanation of what is even being said in the first place. I think when most people understand what is being claimed by "1 = 0.999...", they will immediately see it as a boringly true statement.
Which is exactly how mathematicians see it.
And if you think it is in any way an exciting, provocative statement, which you want to argue against -- then you're probably just not understanding what it claims.