There are 3 types of convergent real number sequences, one that converges to a sequence in the real numbers, one that "converges" to infinity, and one that "converges" to minus infinity?
Obviously only one of the above can happen, if one happens. But does the limsup and the lim inf give all the possibilities of what can happen if it does not converge?
We have that a sequence converges in $\mathbb{R}$ iff limsup = lim inf, and this holds for infinity and minus infinity also, the series converges to infinity iff limsup=lim inf=$\infty$, and converges to minus infinity iff lim sup = lim inf=$-\infty$.
So a sequence diverges in the extended real number system iff lim sup $\ne$ lim inf? And if it does not converge there are only 4 things that can happen?
- limsup $\ne$ lim inf, but both are real, so the sequence ossiciliates between to real numbers?
- limsup $\ne$ lim inf, but lim sup is $\infty$, and lim inf is real, so you can get as high as you want as n increases but you always get down again to a real number.
- The opposite case, where you can get as low as you want as n increases, but you must always get back up again to a real number.
4, limsup = $\infty$ , lim inf $-\infty$, so it osscilliates between all numbers no matter how high n is
(5). if limsup = $-\infty$, or lim inf=$\infty$ we must have convergence to $-\infty$ or $+\infty$.
So, short question: Are these 7 cases the only cases that can happen with any sequence of real numbers?, so a divergent sequence must ossciliate between two fixed numbers?(if we look at infinity as a point it can converge to, or can osscilliate between)?