Mathematics Exam > Mathematics Questions > Consider the following statementa)every conve...
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Consider the following statement
- a)every convergent sequence is a Cauchy sequence
- b)every Cauchy sequence of rational number is convergent
- c)every Cauchy sequence of real number is convergent
- d)every convergent sequence is bounded
Correct answer is option 'C'. Can you explain this answer?
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Consider the following statementa)every convergent sequence is a Cauch...
Explanation:
In order to determine the correct option, let's analyze each statement one by one.
a) Every convergent sequence is a Cauchy sequence:
A convergent sequence is one in which the terms of the sequence approach a specific limit as the sequence progresses. It means that for any given positive value ε, there exists a positive integer N such that for all n ≥ N, the distance between the terms of the sequence and the limit is less than ε.
On the other hand, a Cauchy sequence is a sequence in which the terms become arbitrarily close to each other as the sequence progresses. It means that for any given positive value ε, there exists a positive integer N such that for all n, m ≥ N, the distance between the terms of the sequence is less than ε.
While it is true that every convergent sequence is a Cauchy sequence, the converse is not true. There are Cauchy sequences that do not converge. Therefore, option a) is not correct.
b) Every Cauchy sequence of rational numbers is convergent:
This statement is false. There exist Cauchy sequences of rational numbers that do not converge to a rational number. An example of such a sequence is the sequence of decimal approximations of the square root of 2. This sequence is a Cauchy sequence of rational numbers, but it does not converge to a rational number. Therefore, option b) is not correct.
c) Every Cauchy sequence of real numbers is convergent:
This statement is true. In the real number system, every Cauchy sequence is guaranteed to converge to a real number. This property is known as the completeness of the real numbers. The completeness of the real numbers distinguishes them from the rational numbers. Therefore, option c) is correct.
d) Every convergent sequence is bounded:
This statement is also true. A convergent sequence is bounded because there exists a finite interval around the limit of the sequence that contains all the terms of the sequence. This interval serves as a bound for the sequence. Therefore, option d) is correct.
To summarize, the correct option is c) "every Cauchy sequence of real numbers is convergent".
In order to determine the correct option, let's analyze each statement one by one.
a) Every convergent sequence is a Cauchy sequence:
A convergent sequence is one in which the terms of the sequence approach a specific limit as the sequence progresses. It means that for any given positive value ε, there exists a positive integer N such that for all n ≥ N, the distance between the terms of the sequence and the limit is less than ε.
On the other hand, a Cauchy sequence is a sequence in which the terms become arbitrarily close to each other as the sequence progresses. It means that for any given positive value ε, there exists a positive integer N such that for all n, m ≥ N, the distance between the terms of the sequence is less than ε.
While it is true that every convergent sequence is a Cauchy sequence, the converse is not true. There are Cauchy sequences that do not converge. Therefore, option a) is not correct.
b) Every Cauchy sequence of rational numbers is convergent:
This statement is false. There exist Cauchy sequences of rational numbers that do not converge to a rational number. An example of such a sequence is the sequence of decimal approximations of the square root of 2. This sequence is a Cauchy sequence of rational numbers, but it does not converge to a rational number. Therefore, option b) is not correct.
c) Every Cauchy sequence of real numbers is convergent:
This statement is true. In the real number system, every Cauchy sequence is guaranteed to converge to a real number. This property is known as the completeness of the real numbers. The completeness of the real numbers distinguishes them from the rational numbers. Therefore, option c) is correct.
d) Every convergent sequence is bounded:
This statement is also true. A convergent sequence is bounded because there exists a finite interval around the limit of the sequence that contains all the terms of the sequence. This interval serves as a bound for the sequence. Therefore, option d) is correct.
To summarize, the correct option is c) "every Cauchy sequence of real numbers is convergent".
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Consider the following statementa)every convergent sequence is a Cauch...
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Consider the following statementa)every convergent sequence is a Cauchy sequenceb)every Cauchy sequence of rational number is convergentc)every Cauchy sequence of real number is convergentd)every convergent sequence is boundedCorrect answer is option 'C'. Can you explain this answer?
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