Let ${y_j}_{j=1}^N$ be N given real numbers. Construct a sequence ${a_n}$ so that ${y_j}_{j=1}^N$ is the set of limit points of ${a_n}$, but $a_n \ne y_j$ for any n or j.
My work is as follows, but it's probably wrong: Suppose $a_n = \frac{1}{n}$. Zero is the only limit point, but $0 \notin a_n = \frac{1}{n} \forall n.$
Thank you for your help!
As I understand you have $N$ given points so you can't just decide there is only one point which his zero. You need to work with the specific points that are given. I'll give a hint. Let $m=min\{|y_i-y_j|: 1\leq i<j\leq N\}$. Check that $m$ is well defined and is a positive number. So there exists $n_0\in\mathbb{N}$ such that $\frac{1}{n_0}<m$. Of course it follows that $\frac{1}{n}<m$ for all $n\geq n_0$. So now use these facts and try to build a sequence.