I'm interested in this question apparently posed by John Nash, which I found in the book A Beautiful Mind.
If you make up a bunch of fractions of pi $3.141592\ldots$. If you start from the decimal point, take the first digit, and place decimal point to the left, you get $.1$
Then take the next 2 digits $.41$
Then take the next 3 digits $.592$
You get a sequence of fractions between $0$ and $1$.
What are the limit points of this set of numbers?
I would like to know what can be deduced by assuming $\pi$ is normal?
It seems to me that, if $\pi$ is normal, the sequence defined above must be non-convergent and therefore have at least two limit points (see the discussion here).
But I think by choosing carefully which subsequences to look for, you should be able to construct as many limit points as you like (again, if $\pi$ is normal). For example, you could look for subsequences bounded by $[0,1/3)$, $[1/3,2/3)$ and $[2/3,1]$ to find three distinct limit points.
Could anyone tell me if this reasoning is correct?