Circular Reasoning( Fractions).This kind of proof for 0.99... = 1 tries to show that a = b by starting with the assumption that a/n = b/n. Essentially, it assumes( no proof)  a/n = b/n, multiplies both sides by n, and then cancels n to conclude that a = b.

For example 1/3=.33...   => 1/3*3=.33...* 3 => 1 = .99...

Interestingly, many who present this proof acknowledge its flaws but still claim it as a good proof because it often convinces people.

You can see that most people accept the algebraic proof  below as fact, as evidenced by a simple google search and the overwhelming citations in 'The Naked Emperor' playlist. However, mathematically, this proof has no merit ( as apparent below )  and is just an illusion to believe. Many people stop at this point and delve no further, making it inherently impossible to explain the more complex topic.