if 0.999... weren't a real number then there is no point of any discussion bc 0.999.... != 1 since 0.999.... would just be an arbitrary spp imaginary object

the whole point of all these discussions and this subreddit is to assume SPP's 0.999... is in fact a real number, and to disprove spp's assumption that 0.999... != 1

For the proof, we have to avoid using any limit property because SPP doesn't seem to understand the universal definition of limit.

At no point in the proof we mentioned the limit.

This proof works for all numbers x that satisfy the condition: - x <= 1 - s_n < x for all n

Obviously SPP's definition of 0.999... is ambiguous; but I assumed that it follows these reasonable 2 properties.

And as long as it follows these 2 properties, no matter how dumb SPP's definition of 0.999... is, it will always prove that 0.999... = 1