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

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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