What does it mean if a sequence is not eventually constant From What I understand a sequence $(a_k)$ is eventually constant such that $a_k=a$ whenever $k&gt;N$. Does that mean a sequence is not eventually constant such that $a_k=a$ whenever $k&lt;N$?
No, it means that the points keep moving around.  No matter what point $a$ and number $N$ you pick there is a point $a_k$ with $k \gt N$ and $a_k \neq a$
Yes, it means that for any k, there exist m and n> k such that $a_k\ne a_m$.