I agree that a careful thinker confronted with this puzzle for the first time should eventually conclude that the crux is what exactly the expression "0.999..." actually means. At this point, if you don't know enough math to give a rigorous definition, I think a reasonable response is "I thought I knew what it meant to have an infinite number of 9s after the decimal point, but maybe I don't, and absent me actually learning the requisite math to make sense of that expression I'm just going to be agnostic about its value."

Here's an argument in favor of doing that. Consider the following proof, nearly identical to the one you present. Let's consider the number x = ...999; in other words, now we have infinitely many 9s to the *left* of the decimal point. What is this number? Well,

10x = ...9990

x - 10x = 9

-9x = 9

x = -1.

There are a couple of reasonable responses you could have to this argument. Two of them require knowing some math: one is enough math to explain why the expression ...999 describes the limit of a sequence of numbers that has no limit, and one is knowing even more math than that, so you can explain in what sense it *does* have a limit (the details here resemble the details of 1 + 2 + 3 + ... but are technically easier). I think in the absence of the requisite math knowledge, seeing this argument side by side with the original one makes a pretty strong case for "stay agnostic about whether this notation is meaningful."

And on the third hand, I can't resist saying one more thing about infinite sequences of decimals to the left. Consider the following sequence of computations:

5^2 = 25

25^2 = 625

625^2 = 390625

0625^2 = 390625

90625^2 = 8212890625

890625^2 = 793212890625

It sure looks like there is an infinite decimal going to the left, x, with the property that x^2 = x, and which ends ...890625. Do you agree? Can you find, say, 6 more of its digits, assuming it exists? What's up with that? Is there another x with this property? (Please don't spoil the answer if you know what's going on here without some kind of spoiler warning or e.g. rot13.)

*14 points [-]