Limit of $0.n$ as $n$ approaches infinity.
It's been a while I've been thinking about this question: How does a number change? How 1 reaches 2 for example? Aren't there infinite numbers between these two? This may have more to do with the whole and the part subject in logic but I'd prefer mathematical approach; till I reached to this proof for 0.999 (infinite nines) is equal to 1:Proof for $0.999...=1$
The problem I faced is that the proof supposes 9 for n and continues. Supposing any number for n only makes the growth speed different. What could be the general answer for the limit of 0.n as n approaches infinity?
I'm really curious if the answer could prepare an analytic approach for the first question as well.
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$\begingroup$It is undefined. A way of seeing this is by noting that the subsequence $0.(10^k - 1 ) = 0.999\dots9$ converges to 1, while the subsequence $0.(10^k) = 0.1$ is a constant subsequence converging to $0.1$. Hence the limit does not exist.
$\endgroup$ $\begingroup$Your example is wrong.
How 1 reaches 2 for example?
You are thinking that 1 is reaching 2 like 0.999… reaches 1. You could think that 0.9 is reaching 1 (since they are close enough and the separation between them is nearly 0) but you can't think that there will be 2 after 1 . There are a lot more numbers between them, like 1.0001.
In the case of natural numbers you can think 2 comes after 1.
$\endgroup$ 2 $\begingroup$Let $ x = 0.999999....$
then $10x = 9.99999....$
then, $$10x = 9.999...$$ $$ - x = 0.999...$$ $$9x= 9.000...$$ $$x=1$$
See the Hilbert Hotel for an illustration of why we can pull a 9 across to the units in 10x, and never have any missing 9 in the decimals
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