Example of bounded sequence on $ \mathbb{R}$ s.t. the set $\{X_n : n \in \mathbb{N}\}$ has exactly one accumulation point but $X_n$ is not convergent
My thoughts: Since Xn is bounded it has convergent subsequence. So Xn must be a branch sequence. We can take on branch for instance X_2n=1(so there’s our bounded and convergent subsequence) and for the other part, for instance: X_2n-1 we need one bounded but not convergent subsequence which also has no accumulation points. That’s the part I’m stuck. Could I maybe take something which goes back and forth in the unit interval? Or maybe a series which diverges?(but then how could it be bounded?)
