The simplest "proof"

The simplest proof is as follows: 1/3 = 0.333 ... and both sides are multiplied by 3, 1 = 0.999... 1998, Fred Richman's article "0.999 ... equals to 1"? "This proof is so convincing because people take it for granted that the first step is right, because the equation of the first step is taught like this since childhood." Professor David Tall also found in the survey that many students will turn to doubt the correctness of the first equation after reading this proof. If you think about it carefully, you will find that "1/3 equals 0.333…" and "1 equals 0.999…" are actually the same and unacceptable. Just as many people think that "0.999… can only get closer and closer to 1 but not completely equal to 1", the controversy of "0.333… infinitely close but not equal to 1/3" still exists. The problem has not been solved.

Another controversial piece of evidence.

David Foster Wallace introduced another famous proof in his book "Everything is not just":

Let x = 0.999 ...

So 10x = 9.999. ...

Subtract two expressions, 9x = 9.

So x = 1.

William Byers commented on this proof in How Mathematicians Think: "0.999 ... can represent both the process of adding infinite fractions and the result of this process. Many students only value 0.999 ... as a process, but 1 is a number. How can a process be equal to a number? This is the fuzziness in mathematics. They didn't find that this infinite process can actually be understood as a number. Those students who think that the equation holds after reading the above proof may not really understand the meaning of infinite decimal, let alone understand the meaning of this equation. "