To illustrate the (power of) Bolzano-Weierstrass theorem I am searching for an example of a bounded but not convergent sequence and an explicit convergent subsequence.
I would like it to be non trivial in the sense that (cyclic) sequences like $(-1)^n$ or $1,2,3, 1,2,3, 1,2,3$ or $1,2,3,2,1,2,3,2,1$, where one can easily "see" the (constant) convergent subsequence don't count.
I wanted to use $\sin(n)$, but the construction of a convergent subsequence isn't very explicit.
Best Answer
You might try the sequence $$ 0, 1, \frac{1}{2}, \frac{1}{3}, \frac{2}{3}, \frac{1}{4}, \frac{3}{4}, \frac{1}{5}, \ldots $$ enumerating all rationals in $[0,1]$, which has all sorts of interesting convergent subsequences.