So, lets take .9, .99, .999 and so on. If a sequence of rational numbers converges, it converges to a real number. What number does .9, .99, .999, .9999 (and so on) converge to? Which is to say, is there a number that it gets closer and closer to at every step? Clearly it gets closer and closer to 1 at every step, so the sequence converges to 1.
I don’t think this reasoning is correct, because the limit of an expression does not have to be the same as the value at the point the limit is being taken. For example, if your sequence is “sin(1/x) * x“, then it’ll slowly appear to converge to one as x approaches infinity, but it cannot reach it. So there’s really no relevant conclusion you can make.
You're taking a map from R -> R, I'm taking a sequence of discrete values. It's not the same thing.
However, I was a bit fast and loose, the sequence .9, .99, .999, .9999 also gets closer and closer to 2 but it doesn't converge to 2, I should have said if you have a metric || and some number X such that for any d, there exists an N such that |X -An| < d for all n > N, then the sequence A converges to X. But I wasn't trying to write a proof.
This is one of the many (equivalent) ways the real numbers are defined to begin with, https://en.wikipedia.org/wiki/Construction_of_the_real_numbe...
There are lots of other ways to define sets with operations, but they won't be anything at all like the normal numbers you are used to.