Convolutions in Informal Math

A mathematics instructor makes an attempt to explain why 0.999… = 1 in their blog, and tackles some of the classic explanations as well as many arguments in the comments [via Clicked]. What interested me most was that the writer was frustrated that people can’t accept the arguments, buit buries the real proof of this fact at the end. Instead of laying out from the start the question of what does it mean to say that a repeating decimal is equal to an integer, point out that it has to do with computing a limit, and going from there, the explanation starts with multiplying x=0.9… by 10 to get 10x = 9.9… and subtracting the one equation from the other to find that x=1.
This approach feel vaguely like the kind of argument that actually leads people not to believe the fact is true – it feels like a “trick”. And there is a catch-22 in play. On the one hand, we have the desire to educate people with an innacurate mathematical intuition. However, if we show them the proof involving a limit, there is the real risk that they will zone out, feeling that they don’t “get” math and limits are hard and so on. So, one falls back on an “intuitive” argument, or in fact here a number of different intuitive arguments. And I suspect the average person feels that sure, these magical calculations show that 0.9…=1, but probably some other magical calculations show that they aren’t the same, and there is no real convincing happening, in addition to people becoming even more cynical about “numbers lying”.
Which, I guess my point is that I kind of like the Ask Dr. Math approach (linked in the comments of the original post) which just tackles the limit proof head on, in fairly clear terms I think. It would be nice if it said more about a repeating decimal equaling the limit than just it “is understood to mean”, because that is the crux of the problem – first understand what it means for the equation to be true or false,and then the truth or falseness falls into place fairly quickly.