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The YouTube video titled "Every PROOF you've seen that .999... = 1 is WRONG" discusses the various proofs that have been put forth to show that .999 repeating is equal to 1. The video points out that while the steps in these proofs may be valid, the reasoning used to justify them is often not explicit. The video then goes on to introduce the concept of a limit and use it to demonstrate that the .999 repeating sequence does indeed equal 1.

**00:00:00**In a proof of .999 repeating equal to 1, the steps may be valid, but the reasoning used to justify them is not always explicit. James Murphy introduces an example of a similar expression, 9 repeating and the point existing, which leads to the conclusion that x is minus 1. This implies that all of the other statements in the proof are true, but the proof does not show that .999 repeating equals 1. To show this, Murphy introduces the concept of a limit, which is a concept well known in calculus but not always explained in proofs of algebraic mathematics. By using the fact that 9 multiplying by 10 is the same as moving the decimal point one place to the right, he demonstrates that the proof is valid. He then goes on to introduce the concept ofimplies, which allows for the simplification of proofs without sacrificing the validity of the steps. By hiding the details in another concept that the viewer may not understand, he creates a difficult proof for those who are not familiar with calculus.**00:05:00**The confusion surrounding the definition of .999 repeating comes from the fact that it is not a mystical number with an infinitely many nines after the decimal point. .99 repeating is simply the number that this sequence approaches, if there is such a number. In mathematical terms, it is the limit of the sequence. Finally, a proof is given that shows that the sequence is getting closer and closer to 1.

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