What does it mean if a sequence is not eventually constant

Question

From What I understand a sequence $(a_k)$ is eventually constant such that $a_k=a$ whenever $k>N$. Does that mean a sequence is not eventually constant such that $a_k=a$ whenever $k<N$?

Answer 1

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 $a_k \neq a$

Answer 2

Yes, it means that for any k, there exist m and n> k such that $a_k\ne a_m$.

Answer 3

Think what this means in english.

A sequence is eventually constant means an some point all the values from than on out are going to be the same.

That is to say the sequence is $\{a_1, a_2, a_3,........, a_{n-1}, a_n, a,a,a,a,........,a,a,a,a,a,....\}$ where after the $n$ term all the remaining terms are the same.

A sequence not being eventually constant means ... that that isn't the case; that no mater how far you go sequence's term will vary and not settle to a single value.

That the seqence is $\{a_1, a_2, a_3,........, a_{n-1}, a_n, a_{n+1}, a_{n+2}....\}$ and that it isn't the case that all the $a_i$ are the same after some point. That's really all. It's not a complicated concept.

In math we say:

${a_k}$ is eventually constant if there is some $N$ and some value $a$ so that for all $k > N$, $a_k = a$.

In other words, we the sequence is $\{a_1, a_2, ....., a_{N-1}, a_N, a,a,a,a,a.....\}$

If that is not true that means for any $a_k$ there will always be a $m > k$ where $a_m \ne a_k$ (and then there will but an $n > m$ where $a_n \ne a_m$ and so on.

No matter how far you go, there will always be terms further along that are not equal to each other.

Answer 4

A sequence $(a_n)_{n\geq0}$ is not eventually constant iff for all $N$ there are $j$, $k>N$ with $a_j\ne a_k$.

No mention of a particular $a$ here.