Choose the correct statements.a)Every infinite and bounded set must have a limit pointb)Any finite set cannot have a limit pointc)Any infinite but unbounded set cannot have a limit pointd)Every closed and bounded set has a limit pointCorrect answer is option 'A,B'. Can you explain this answer?

Question

Answer

Bolzano weierstrass theoremstate that, every bounded infinite set ofreal numbers has at least one limit point.

So (A) is true satetement.

Also if x be limit point of any set A, then by definition evely neighbourhood of x must contain infinitely many points of A, which is not possible if A is finite. So (B) is true statement.

(C) is wrong, as

i.e. A have limit points.

(D) is wrong, as singleton set are always closed and bounded in ℝ. But do not have limit point.

Answer

Understanding Limit Points
In set theory, limit points (or accumulation points) are crucial in understanding the behavior of sets, especially in analysis. Let’s analyze each statement:
Statement A: Every infinite and bounded set must have a limit point
- True: By the Bolzano-Weierstrass theorem, every infinite bounded subset of real numbers has at least one limit point. For example, the set of points in the interval (0,1) includes sequences that converge to points in that interval, thus guaranteeing limit points.
Statement B: Any finite set cannot have a limit point
- True: A finite set consists of isolated points, meaning there are no points that can be arbitrarily close to each other as required for a limit point. No matter how close you get to any point in a finite set, you will never find another point of the set nearby.
Statement C: Any infinite but unbounded set cannot have a limit point
- False: An infinite unbounded set can have limit points. For instance, the set of all real numbers can be considered unbounded and does not have a limit point, but a set like the integers, while unbounded, has limit points in the context of sequences converging to limits in the real line.
Statement D: Every closed and bounded set has a limit point
- False: While every closed and bounded set in the real numbers contains all its limit points, it does not guarantee that it has additional limit points. For example, the closed set {0} has no limit points but is closed and bounded.
Conclusion
Thus, the correct answers are options A and B. Statements A and B are valid based on the definitions and properties of limit points, while C and D do not hold true in every context.

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