I'd like to expand on one key point made in other threads: in mathematics it's important that we agree on the rules first, otherwise we're just going to be talking past one another.
When we make a mathematical statement like 0.999... = 1, in a literal sense that is just a series of ascii symbols. It is neither 'true' nor 'false' in a meaningful way, unless we make concrete what we mean by the parts of it. When most mathematicians / students see '0.999...' with no further clarification, make a couple of leaps about what is meant there.
First of all, that we are defining a 'real' number, constructed in one of the usual ways. As an aside - the real numbers aren't really 'real'. In mathematics


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