Hi guys:
the Qs is 'Find the superior and inferior limit of the sequence"(-1)^n(1+n^-1)". Even I know the answer for the Qs is 1 and -1, but what should be the currect process for this Qs, please give some detail.thanks!! <img smilieid="84" class="inlineimg" src="http://www.mathhelpforum.com/math-help/images/smilies/Skype/emoticon-0139-bow.gif" border="0">
Your sequence is given by . Calculate a few values: , , , .
In fact it should be clear that . If n is odd that is which converges to -1 and if n is even that is which converges to 1. Since, if a sequence converges, every convergent subsequence must converge to the same thing, it is clear that every convergent subsequence converges to either 1 or -1.
Now, "lim sup" is defined as the supremum (least upper bound) of the set of subsequential limits while the "lim inf" is the infimum. Here, the set of subsequential limits is {-1, 1} which has minimum -1 and maximum, 1 and, since it has a minimum and maximum, those are the "inf" and "sup" and so the lim inf and lim sup.