let {a n } be a convergent sequence of cluster points of {x n } and let the lim

Question

let {an} be a convergent sequence of cluster points of {xn} and let the lim an as n approaches equal a . prove that a is also a cluster point of {xn}.

Explanation / Answer

In this case, a sequence of sets approaches a limiting set when the elements of each member of the sequence approach the elements of the limiting set. In particular, if {Xn} is a sequence of subsets of X, then: lim sup Xn, which is also called the outer limit, consists of those elements which are limits of points in Xn taken from (countably) infinitely many n. That is, x ? lim sup Xn if and only if there exists a sequence of points xk and a subsequence {Xnk} of {Xn} such that xk ? Xnk and xk ? x as k ? 8. lim inf Xn, which is also called the inner limit, consists of those elements which are limits of points in Xn for all but finitely many n (i.e., cofinitely many n). That is, x ? lim inf Xn if and only if there exists a sequence of points {xk} such that xk ? Xk and xk ? x as k ? 8. The limit lim Xn exists if and only if lim inf Xn and lim sup Xn agree, in which case lim Xn = lim sup Xn = lim inf Xn Special case: discrete metric In this case, which is frequently used in measure theory, a sequence of sets approaches a limiting set when the limiting set includes elements from each of the members of the sequence. That is, this case specializes the first case when the topology on set X is induced from the discrete metric. For points x ? X and y ? X, the discrete metric is defined by So a sequence of points {xk} converges to point x ? X if and only if xk = x for all but finitely many k. The following definition is the result of applying this metric to the general definition above. If {Xn} is a sequence of subsets of X, then: lim sup Xn consists of elements of X which belong to Xn for (countably) infinitely many values of n. That is, x ? lim sup Xn if and only if there exists a subsequence {Xnk} of {Xn} such that x ? Xnk for all k. lim inf Xn consists of elements of X which belong to Xn for all but finitely many n (i.e., for cofinitely many n). That is, x ? lim inf Xn if and only if there exists some m>0 such that x ? Xn for all n>m. The limit lim X exists if and only if lim inf X and lim sup X agree, in which case lim X = lim sup X = lim inf X.[2] This definition of the inferior and superior limits is relatively strong because it requires that the elements of the extreme limits also be elements of each of the sets of the sequence. Using the standard parlance of set theory, consider the infimum of a sequence of sets. The infimum is a greatest lower bound or meet of a set. In the case of a sequence of sets, the sequence constituents meet at an set that is somehow smaller than each constituent set. Set inclusion to provides an ordering that allows set intersection to generate a greatest lower bound nXn of sets in the sequence {Xn}. Similarly, the supremum, which is the least upper bound or join, of a sequence of sets is the union ?Xn of sets in sequence {Xn}. In this context, the inner limit lim inf Xn is the largest meeting of tails of the sequence, and the outer limit lim sup Xn is the smallest joining of tails of the sequence. Let In be the meet of the nth tail of the sequence. That is, Then Ik ? Ik+1 ? Ik+2 because Ik+1 is the intersection of fewer sets than Ik. In particular, the sequence {Ik} is non-decreasing. So the inner/inferior limit is the least upper bound on this sequence of meets of tails. In particular, So the superior limit acts like a version of the standard supremum that is unaffected by set elements that occur only finitely many times. That is, the superemum limit is a set that is a superset (i.e., an upper bound) for all but finitely many elements. Similarly, let Jm be the join of the mth tail of the sequence. That is, Then Jk ? Jk+1 ? Jk+2 because Jk+1 is the union of fewer sets than Jk. In particular, the sequence {Jk} is non-increasing. So the outer/superior limit is the greatest lower bound on this sequence of joins of tails. In particular, So the inferior limit acts like a version of the standard infimum that is unaffected by set elements that occur only finitely many times. That is, the infimum limit is a set that is a subset (i.e., a lower bound) for all but finitely many elements. The limit lim Xn exists if and only if lim sup Xn=lim inf Xn, and in that case, lim Xn=lim inf Xn=lim sup Xn. In this sense, the sequence has a limit so long as all but finitely many of its elements are equal to the limit. you will get information in this given below link http://en.wikipedia.org/wiki/Limit_superior_and_limit_inferior

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