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Does Champernowne's constant converge to the digits of $\pi$?
Clearly if you go far enough into Champernowne's constant you will find a string of the first $n$ digits of $\pi$ for any finite $n$.
For every finite set of $n$ digits of $\pi$ contained within the constant, $n+1$ digits are subsequently represented $a(n+1)$ digits later, for some $a\leq b$ where $b$ is the base.
By induction there is no string of digits which is not, ultimately, represented.
If it contains the digits of $\pi$, it cannot contain anything after them since they never end, so if it contains them, it must contain them at its end. Could it converge to the digits of $\pi$?
However it would seem we can never reach them if it did, since they would occur at a length greater than the length of the digits of $\pi$. Does this mean they are not represented?