Not assuming the LOPS 'limit of partial sums' to be the 'sum' of an 'Infinite Series' (Instead, 'S' = an 'applied pattern').

We're going to prove that 1 = 0.999..., but in a raw (sensible) way. I.S. (Infinite Series): 1 - 0.999... = 1 + (-0.999...) = 1 + (-0.9) + (-0.09) + (-0.009) + ...

s1 = 1

s2 = 0.1

s3 = 0.01

s4 = 0.001

Since my 'S' = 'applied pattern' (pattern from partial sums), 'S' = 0.000...1. (an increasing # of 0's followed by a 1). Now to see if this number has any positive value, we want to find a characteristic difference between the 'zeroes' and the '1.' And there is a difference: ALL the 'zeroes' hold an INTEGER POSITION, as the '1' does not. Note that a decimal expansion is just a sequence (A sequence of digits). Since there is no largest integer, this means there is no 'infinith' (final) 0. This means there can't be a '1' either since the '1' would come after said zero.

So, the '1' is there due to a pattern, but then we rid of it when we see that it does not maintain an 'integer position.' That's the commonality of a finite and infinite sequence [In both, all terms must be a (positive integer) term]. For example, in 0.999..., there is no 'infinith' 9. ALL the 9's hold an integer position. So, 0.000...1 is actually 'absolute 0' since the '1' is not sequenced. You can't have a term such as X∞ in an infinite sequence because this would imply that ∞ is AN INTEGER, as a position can't be, for example, the 5.5th term. ∞ being an integer makes no sense because why can't (0.5 + 1 + 1 + 1 + ...) be ∞. This is not an integer. So, 0.000...1 = 0.000... = 0. Based on the math axiom: If X - Y = 0, then X = Y, we have proven that 0.999... = 1, and that the LOPS is the 'true' sum of an I.S.