Cycle with infinite length on $[0,1]$?

Question

Suppose that every point in a unit interval $[0,1]$ is a node, and consider directed edges which can connect any point in unit interval to another.

I wonder if we can find any infinite cycle. I tried to find one using a convergent sequence $\{s_n\}$ converging toward $s^\infty$, where $s_n\rightarrow s_{n+1}$ and $s_\infty\rightarrow s_1$, but this can't be right because the sequence never reaches $s_\infty$.

Any idea on this? or examples?

Answer 1

I wonder if we can find any infinite cycle. I tried to find one using a convergent sequence $\{s_n\}$ converging toward $s^\infty$, where $s_n\rightarrow s_{n+1}$ and $s_\infty\rightarrow s_1$, but this can't be right because the sequence never reaches $s_\infty$.

And this holds true for any "cycle with infinite period". You'll never reach the link connecting back to the start to make a cycle. Either you accept the concept of infinite ordinals as a valid solution, or you do not. Based on that decision you either have many solutions, or you don't have any at all.

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It is however possible to make directed edges such that you can find any cycle of arbitrary finite length. It's trivial in fact. Connect every number to $1$, and connect $x$ to $x/2$. Now starting at $1$ you can have an arbitrarily long cycle.