Letting n = 1, 2, 3, ... in succession, we find that B consists of the points -1/2, 2/3, -3/4, 4/5, -5/6, 6/7, ...

The sequence of positive points (which consists of all the points for even values of n) converges to +1, and the sequence of negative points (which consists of all the points for odd values of n) converges to -1, so the only limit points of B are +1 and -1.

(b) Is B a closed set?

B is not a closed set, because NEITHER of the LIMIT POINTS of B is an ELEMENT of B.

To see this, first set [(-1)^n](n)/(n + 1) to +1, and solve for n. That gives [(-1)^n](n) = n + 1.

For all even (and ...

Solution Summary

The limit points of B and the closure of B were found. Also, it was determined whether B is open, whether B is closed, and whether B contains any isolated points. A detailed explanation was given for each part of the solution.

Consider the following subsets of (FUNCTION1) and (FUNCTION2). The subspaces X and Y of (SYMBOL) inherit the subspace topology. In the following cases determine the interior, the closure, the boundary and the limitpoints of the subsets:
1, 2 and 3
*(For complete problem, including properly cited functions and symbols, pleas

1) Let M be an elementary set. Prove that | closure(M)M | = 0. (closure of M can also be written as M bar, and it is the union of M and limitpoints of M).
2) If M and N are elementary sets then show that
| M union N | + | M intersection N| = |M| + |N|
The definition of elementary set : If M is a union of finite members

Let Fr(A) denote the frontier set of A and Cl(A) denote the closure of A, where A is a subset of R^n. Solve the following problems.
Exercise 2.6: For any set A, Fr(A) is closed.
Exercise 2.12: If A and B are any sets, prove that Cl(A and B) belongs to Cl(A) and Cl(B). Give an example where Cl(A and B) is empty, but Cl(A) a

See Attached
If A is any subset of let denote the collection of all closed sets containing A. The set is called the closure of A.Note that is a closed set, prove that it is the smallest closed set containing A.Prove the following

Let Y be a subspace of X and let A be a subset of Y. Denote by Cl(A_X) the closure of A in the topological space X and by Cl(A_Y) the closure of A in the topological space Y. Prove that Cl(A_Y) is a subset of Cl(A_X) . Show that in general Cl(A_Y) not equal Cl(A_X). See the attached file.

Prove that the point p is a limit point of the point set X if and only if each open point set containing p contains a point in X which is different from p. Prove without using sequences. Only use the def. of open set, open interval, and that the point p is a limit point of the point set X means that each open interval containing

Question: Construct a bounded set of real numbers with exactly three limitpoints (put the limitpoints at 0, 1 and 2005).
(Please explain in your own words how the proof works. If you use a theorem, please state what it is and if possible, where you got it).