I found a statement in a book concerning the decimal expansion of $\pi$ that I do not really understand. The statement is my problem number 2, where problem number 1 really looks like a reference request. Here there is the setting.
Take $\pi$, and construct a sequence $x_1 := 0.1$, $x_2 := 0.41$, $x_3 := 0.592$, etc, where for example the first element of the sequence is the first digit of the decimal expansion of $\pi$, and the second element of the sequence corresponds to the second and the third digit of the decimal expansion of $\pi$.
1. Apparently, it has been proved somewhere that this sequence converges at least to a limit point. I would like to know if somebody knows about this result.
2. I do not really get how this sequence, if it converge (as it has been proved), can converge to more than one point.
Is there somebody who can clarify point 2?
Thank you in advance.
The sequence is contained in $[0,1]$, which is a compact set. Sequences in a compact set must have at least an accumulation point. In other words, there is a convergent subsequence.
A sequence can have more than one accumulation point. For example:
$$0.1,0.6,0.01,0.51,0.001,0.501,0.0001,0.5001,\ldots$$