I don't think that is true.
Consider the sequence
and the sequence
so is
Now
I am trying to prove the following:
.
Here's what I have so far:
FTSOC assume . We know that and for only finitely many . Then for only finitely many . This implies almost always. So, almost always.
But this doesn't get me anywhere. Where am I going wrong? If I let , I just get . So, this tells me nothing.
Any help would be appreciated. (Even though it doesn't state it, I am assuming that all are nonnegative.)
For only nonnegative sequences, Drexel28 is giving you the hint to only look at the inf of the set of values at any subsequence (ignore the order of the sequence).
Let .
Consider any element and , (by definition since everything is nonnegative).
Also, remember that the inf of a subset of AB must be larger than the inf of AB