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 $k<N$?
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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 $a_k \neq a$
Ross Millikan
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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.
Christian Blatter
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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.
fleablood
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Does that mean if a sequence is not eventually constant, it will always be divergent? – S A Feb 23 '18 at 14:08
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No. Take the sequence $\frac 12, \frac 23,\frac 34, \frac 45,....., \frac {n}{n+1}$. No two terms are the same so it is not eventually constant. But it converges to $1$. (Which BTW absolutely no terms actually equal). Eventually constant means there comes a point were every term from then all is the same. Not eventually constant simply means that doesn't happen. – fleablood Feb 23 '18 at 16:37
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I now understand, thank you :) – S A Feb 23 '18 at 22:19
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Yes, it means that for any k, there exist m and n> k such that $a_k\ne a_m$.
user247327
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How can that possible be a "yes". "for any k, there exist m and n> k such that ak≠am" is completely different than "there is an $n$ so that all $a_k = a$ if $k < N$" – fleablood Feb 23 '18 at 01:33