To recap:
The reason this proof is wrong is because the two underlined numbers are not equal and they are not equal because the number of nines in the first is lower than the number of nines in the second.
An image representing (10 \times 0.\dot9) operation:
The green rectangle (representing the first number) is shorter than the purple rectangle (representing the second number).
This is based on the premise that (0.\dot9) represents a specific infinite quantity. However, most people do not see it this way. So it might not be the best counter-argument.
So let us define (0.\dot9) to be a non-specific infinite number instead.
A number (x) is said to be specific if (x - x = 0). If (x - x \neq 0) then (x) is not a specific number.
According to some, (\infty - \infty = \infty). This is fine, nothing wrong with it, but let’s see where it leads.
If (\infty - \infty = \infty) then (0.\dot9 - 0.\dot9 \neq 0). This is because the number of non-zero terms in the first sum is not the same as the number of non-zero terms in the second – there’s always an infinite difference. Hence, (10x - x \neq 9x).
Indeed, the same conclusion follows for any case where (\infty - \infty) is non-zero.