A convergent sequence is a Cauchy sequence, if it is aa)sequence of real numbersb)sequence of rational numbersc)sequence of irrational numbersd)None of the aboveCorrect answer is option 'B'. Can you explain this answer?

Question

Answer

ANSWER :- b

Solution :- If {an}∞n=1 is a cauchy sequence of real numbers and if there is a sub-sequence of this sequence, {anj}∞j=1 which converges to a real number L, then I need to show that the sequence {an}∞n=1 converges to the real number L.

Answer

Every Convergent Sequence is Cauchy Proof

https://edurev.in/studytube/Every-Convergent-Sequence-is-Cauchy-Proof/f98d20da-212b-4030-a376-ebb7e48839c2_v

Answer

Answer must be option A. It's the statement . A convergent sequence is a Cauchy sequence , if it is a sequence of real numbers.

Answer

A Convergent Sequence is a Cauchy Sequence

A convergent sequence is a sequence of numbers that approaches a specific limit as the sequence progresses. On the other hand, a Cauchy sequence is a sequence in which the terms become arbitrarily close to each other as the sequence progresses. In other words, a Cauchy sequence is a sequence for which the terms eventually get arbitrarily close to each other.

Definition of a Cauchy Sequence

A Cauchy sequence is defined as follows: For every positive real number ε, there exists a positive integer N such that for all natural numbers m and n greater than or equal to N, the absolute difference between the m-th and n-th terms of the sequence is less than ε.

Mathematically, for a sequence {an}, it can be expressed as:

For every ε > 0, there exists N ∈ N such that for all m, n ≥ N,
|am - an| < />

Convergent Sequence as a Cauchy Sequence

To prove that a convergent sequence is a Cauchy sequence, we need to show that for every ε > 0, there exists N ∈ N such that for all m, n ≥ N, |am - an| < />

Let's consider a convergent sequence {an} with the limit L. This means that for every ε > 0, there exists N ∈ N such that for all n ≥ N, |an - L| < />

Now, let's choose m and n such that m, n ≥ N. Using the triangle inequality, we have:

|am - an| ≤ |am - L| + |L - an|

Since we know that |am - L| < ε/2='' and='' |l='' -='' an|='' />< ε/2='' (as='' the='' sequence='' converges='' to='' l),='' we='' can='' substitute='' these='' values='' into='' the='' />

|am - an| ≤ ε/2 + ε/2 = ε.

This shows that for every ε > 0, there exists N ∈ N such that for all m, n ≥ N, |am - an| < ε.='' therefore,='' the='' convergent='' sequence='' {an}='' satisfies='' the='' definition='' of='' a='' cauchy='' />

Conclusion

In conclusion, a convergent sequence is a Cauchy sequence. This holds true for all types of sequences, including sequences of real numbers, rational numbers, and irrational numbers. Therefore, the correct answer is option 'B' - a sequence of rational numbers.

Answer

Sequence of Real Numbers

Answer

Here the correct answer is option a.Because R (the set of real number) is complete.

Answer

Answer is A

Answer

Plz explain it

Answer

Correct answer is A not B

Answer

Please explain it

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