# Eventually self-outgrowing = eventually monotonic?

• May 31st 2012, 05:38 AM
jens
Eventually self-outgrowing = eventually monotonic?
Hi,

Consider a bounded sequence $\displaystyle (a_n)_{n \in \mathbb{N}}$ whose terms are eventually "outgrown" by other terms of the sequence, i.e., for any index $\displaystyle n$ there exists an $\displaystyle n' > n$ such that $\displaystyle a_{n'} > a_n$. Does this property imply that $\displaystyle a_n$ is eventually monotonic, i.e., that there exists an index $\displaystyle n_0$ such that $\displaystyle a_n$ is monotonically increasing from index $\displaystyle n_0$ onwards? I can't think of any counterexample. If true, do you know if this a known theorem which I can reference from some standard textbook?

NB: the boundedness assumption is essential, otherwise you can construct simple counterexamples like $\displaystyle a_n = n + (-1)^n$

Thanks,
jens
• May 31st 2012, 05:46 AM
girdav
Re: Eventually self-outgrowing = eventually monotonic?
$\displaystyle a_{2n}:=1-\frac 1{2n}$ and $\displaystyle a_{2n+1}=0$.
• May 31st 2012, 02:49 PM
jens
Re: Eventually self-outgrowing = eventually monotonic?
Sorry for the mess, but I had only sequences in mind whose variations become "tiny", that is, whose $\displaystyle \lim\sup$ and $\displaystyle \lim\inf$ coincide. But I forgot to tell. So let me correct my question:
Assume $\displaystyle a_n$ converges to a finite limit and for any index $\displaystyle n$, there exists $\displaystyle n' > n$ such that $\displaystyle a_{n'} > a_n$. Does this imply that there exists an index $\displaystyle n_0$ such that for any two $\displaystyle n_1$ and $\displaystyle n_2$ both larger than $\displaystyle n_0$, we have $\displaystyle n_1 < n_2 \Rightarrow a_{n_1} < a_{n_2}$?
What about $\displaystyle a_{2n}=1-1/n^2$ and $\displaystyle a_{2n+1}=1-1/n$?