I have a sequence $(x_n)$ with a property that $\liminf x_n \in [-1,1]$ and $\limsup x_n \in [-1,1]$. I'd like to express this property using big-O and small-o somehow.
I'm thinking:
$x_n = \gamma + o(1)$ where $\gamma \in [-1,1]$
(I don't think this is right since we don't know if $x_n$ converges to $\gamma$ or not, we only know that $x_n$ fall in $[-1,1]$ for large $n$)
or I can express this as:
$x_n \ge -1 - |o(1)| $ and $x_n \le 1 + |o(1)| $
(I use absolute value to get positive sequence $o(1)$). I've always seen "$= o(1)$" with equality, and never seen inequalities "$ > o(1)$" or "$ < o(1)$", so I'm not sure if this is right.
Do you think (2) is correct or how would to write it?
Thank you!
All you can say in those notations (and if you want $x_n$ alone on the left of the equality sign) is that $$ x_n=O(1)$$ (or equivalently $x_n=c+O(1)$ for arbitrary $c$). We do not have $x_n=o(1)$ because the sequence $x_n=(-1)^n$ would be a valid example with $\frac{x_n}{1}\not \to 0$.
At any rate, the notations "$<o(1)$" or similar are not defined (though one could think of a way to come up with a suitable definition). Recall that even the $=$ in $f(n)=O(g(n)$ is an abuse of notation. A more formal approach would be to consider $O(g(n))$ a set of functions and write $f\in O(g)$.
Also note that adding an absolute value is meaningless (in the problem at hand). Again, recall that the definition of the Landau symbols involves taking absolute values anyway.