Which one of the following is incorrect?a)every bounded sequence has a convergent subsequenceb)every sequence has a monotonic subsequencec)every sequence has a limit pointd)every sequence has a countable number of termsCorrect answer is option 'C'. Can you explain this answer?

Question

Answer

The Incorrect Statement: The incorrect statement is option 'C' which states that every sequence has a limit point.

Explanation:
A sequence is a list of numbers written in a specific order. It can be finite or infinite. A limit point is a point that can be approached arbitrarily closely by an infinite number of terms in a sequence. In other words, it is a point where the terms of the sequence cluster around.

The incorrect statement implies that every sequence has at least one limit point. However, this is not true. There are sequences that do not have any limit points, and therefore, option 'C' is incorrect. Let's understand this in detail.

Counterexample:
Consider the sequence {1, 2, 3, 4, ...} which is an increasing sequence of positive integers. This sequence does not have any limit point.

To prove this, suppose there is a limit point 'x' for this sequence. Since 'x' is a limit point, there must be infinitely many terms of the sequence that are arbitrarily close to 'x'. However, in this case, we can easily find a positive integer 'N' such that all terms of the sequence with indices greater than 'N' are greater than 'x'. Therefore, 'x' cannot be a limit point.

Hence, the sequence {1, 2, 3, 4, ...} is an example of a sequence that does not have any limit point, and thus, option 'C' is incorrect.

Other Options:
- Option 'A' states that every bounded sequence has a convergent subsequence. This statement is true and is known as the Bolzano-Weierstrass theorem.
- Option 'B' states that every sequence has a monotonic subsequence. This statement is also true and can be proved using the Bolzano-Weierstrass theorem.
- Option 'D' states that every sequence has a countable number of terms. This statement is true as a sequence can be finite or infinite, and both cases have countable terms.

In conclusion, option 'C' is the incorrect statement as there are sequences that do not have any limit points.

Answer

The set of all cluster points of a sequence is sometimes called the limit set. contains all but finitely many elements of the sequence). That is why we do not use the term limit point of a sequence as a synonym for accumulation point of the sequence.

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