Why aren't all real numbers equal to one another?

I know, stupid question. But humor me for a sec. First off, we know that all real numbers have two numbers which are infinitely close to them, right? That would seem to be, for any given value of x,

x ± y

where

y = .000...1

But here's the thing:

y + .999... = 1

Right?

And, of course, we all know that .999... = 1, so that means that y = 0, right? Which means that all numbers infinitely close to one another, which represents the entirety of the real number line, are equal, right? Something here is screwed up, but for the life of me I can't figure out what.

PS, I wasn't sure what tag to give this, so feel free to edit them.

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What is $0.000\ldots1$ exactly? – Alqatrkapa Jul 16 '14 at 16:48
@Alqatrkapa Sorry, I wasn't sure how to write that. Infinite 0's followed by a 1, meant to represent an infinitely small number. – KnightOfNi Jul 16 '14 at 16:49
What you describe is not well-defined as a real number. – qaphla Jul 16 '14 at 16:50
Um, all you are showing seems to be $x=x$. Nothing here implies that any 2 real number are the same. – Gina Jul 16 '14 at 16:57
BEGIN QUOTE "First off, we know that all real numbers have two numbers which are infinitely close to them, right?" END QUOTE ${}\quad{}$ No. That is false. ${}\qquad{}$ – Michael Hardy Jul 16 '14 at 18:14

Your mistake is in assuming that there is a real number $y=0.000\ldots1$, but actually there is no such thing. The real number system contains no "infinitely small" elements.

It is also wrong when you assert that

we know that all real numbers have two numbers which are infinitely close to them, right?

Two different real numbers are always a finite distance from each other.

You can get as close to your $x$ as you want without actually hitting it, but that is not the same as saying that you can get "infinitely close" to it.

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Wow, that's an interesting way of looking at it. I suppose "infinitely close" was just a figure of speech my teachers have been using. – KnightOfNi Jul 16 '14 at 16:53
@KnightOfNi: Either that, or they are confused themselves. 200 years ago at the dawn of analysis it was not uncommon to speak of "infinitely small" quantities (Cauchy did it, for example, as a shorthand for limit arguments), but they proved to be difficult to justify in a way that would make sense for students, and today analysis is almost universally taught in terms of explicit limits of finite quantities. – Henning Makholm Jul 16 '14 at 17:00
As an aside to that, it's common for people to say "infinitely close" when what they mean is "arbitrarily close". – qaphla Jul 16 '14 at 17:00
(cue for someone to start shilling non-standard analysis in 3 ... 2 ... 1 ...) – Henning Makholm Jul 16 '14 at 17:06
If you want another example of the difference between "infinitely" and "arbitrarily": You can always find an integer larger than any given integer (arbitrarily large), but there's no integer larger than all others (infinitely large). – Henry Swanson Jul 16 '14 at 18:22

"Infinitely many zeroes followed by a 1" is actually not a well-defined decimal numeral.

The places in a decimal numeral are all indexed by integers that denote their location relative to the unit place: e.g. the hundred's place has index $2$ and the thousandth's place has index $-3$.

If you have infinitely many zeroes to the right of the decimal point, that means every place whose index is a negative integer has to be a zero: so there aren't any places left to put a $1$!

One can create a numeral system that would allow numerals like the one you wrote, but there isn't a good number system for them to correspond to. e.g. what would $0.\overline{0}5 + 0.\overline{0}5$ be?

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