I'm really confused about the concept of infimum and supremum, basicly in order to find infimum and supremum I just start with $n=1$, look at what I get and then I put $n=2$ and look at it too, then I conclude what will be the infimum and supremum.
The problem with that method is that it may be hard when the expression is too complicated and I dont 'formaly' prove the infimum and supremum.
Other concept that drives me crazy is that $\limsup$ and $\liminf$ I dont understand why it differs from the limit of the sequence itself and how to calculate it.
It would be great for some numerical examples and explanations
Thanks in advance
tl;dr
A) What are the steps to calculate the inf/sup of a sequence? for example this sequence $$a_n=(n^2)^{n-1}-n^3+e^n + 4\sqrt(n)$$
B) what does it mean lim inf and lim sup?
Best Answer
If you have a numerical sequence, 5 values can be of interest:
infimum: minimum if is reached, otherwise infimum is the "minimum in the limit", i.e. the greatest lower bound. For instance $1/n$ does not have a minimum, and has infimum $0$
supremum is the symmetrical of infimum.
limit: does not always exist, value from which you can get arbitrarily close from some point on. Formally there is a limit $l$ if for all $\epsilon>0$ there is an index $N$ such that after $N$, all your terms are at a distance less than $\epsilon$ from $l$.
lim inf: minimum of limits of extracted sequences. For instance if you have $(-1)^n$, limits of extracted sequence are $-1$ and $1$, so the lim inf is $-1$ (in this case it is also the minimum). But if your sequence starts with $-2$ and then it is $(-1)^n$, the minimum and the lim inf are different.
lim sup: symmetrical of lim inf.
Exercise: try to prove that when there is no minimum, then infimum=lim inf.