Edited sentence: Although this is a mainstream proof used in schools, it is not correct and fails to take into the countability infinite number of 9s.Another example of why it is so important to not treat infinitesimally small numbers as zero consider the following:.Approaches the limit of 0, but does not actually reach it.For example, I may assert that the limit of 1/x as x tends to infinity, is 0, however there is no number for which 1/x is actually 0... The important thing to learn here is that a limit is an asymptotic boundary, and an asymptote is never reached.If 1/x were to be 0 then this would contradict the assertion that the expression above approached infinity, in accordance to the axioms of Mathematics laid out in Bertrand Russel's Principia Mathematica....If we were to draw a graph of this, initially we see that the expression appears to approach 0 asymptotically, however as we move out in the graph we see it approaches infinity.In Cantor's Mathematics, being able to equate infinities as countably the same size, is crucial to a correct analysis.In this example, we can consider the Taylor series of e^x, and remove the terms which are zeroed by the 1/x term, so we can show that the this term approaches infinity.In this second example, we can see that there is one more 9 following the decimal point for the x than there is in the 10x term, so the two infinite sets of 9s after the decimal point, are not the same size.Just because you show that the limit of 0.999... is 1, does not mean that it is or can ever be.Now, we know that anything multiplied by 0 is 0.The flaw in this proof goes as follows:.The problem with this proof is that it asserts that the limit can actually be the same as the value at a point.Therefore we know that the term:.This draws attention to the importance of not disregarding an infinitesimally small value as merely 0 because it's an inconvenient reality.This is a fundamental axiom of mathematics, and is not open for debate.This is a nice proof attempt, but again, it's wrong