Which of the following statement is/are correct?a)Every bounded sequence an has atleast one limit pointb)Every bounded sequence has no limit pointc)The set of limit points of a bounded sequence an is unboundedd)None of the aboveCorrect answer is option 'A'. Can you explain this answer?

Question

Answer

Statement (a): Every bounded sequence an has at least one limit point

A bounded sequence is a sequence in which all the terms are within a certain range or interval. A limit point is a point that can be approached by an infinite number of terms in the sequence.

Explanation:
To prove statement (a), we need to show that every bounded sequence has at least one limit point.

Let's consider a bounded sequence {an}. Since the sequence is bounded, there exists a number M such that |an| ≤ M for all n, where M is a positive real number.

Now, let's consider the set of terms {an} that are greater than M/2. If this set is empty, then all the terms of the sequence are less than or equal to M/2. In this case, we can choose any number greater than M/2 as a limit point, and that number will be approached by an infinite number of terms in the sequence.

If the set of terms {an} that are greater than M/2 is not empty, then it must have a least upper bound (supremum), which we can denote as L. Since L is the least upper bound, there exists a term an in the sequence such that L - 1/n < an='' ≤='' l='' for='' all='' positive='' integers='' />

As n approaches infinity, L - 1/n approaches L. Therefore, L is a limit point of the sequence.

In either case, we have shown that every bounded sequence {an} has at least one limit point.

Therefore, the correct statement is (a) Every bounded sequence an has at least one limit point.

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