Question

Prove that every cauchy sequence is bounded

Answer 1

Every Cauchy sequence is bounded.

Step-by-step explanation:

Let's prove that every Cauchy sequence is bounded.

1. Assume that we have a Cauchy sequence, {an}.
2. Since it is a Cauchy sequence, for any positive real number ε, there exists a positive integer N such that for all n, m > N, |an - am| < ε.
3. Take ε = 1 and choose N such that |an - am| < 1 for all n, m > N.
4. Now, consider the subsequence {aN+1}, {aN+2}, {aN+3}, ...
5. Since the differences between any two terms in this subsequence will always be less than 1, the subsequence is bounded.
6. Since this subsequence is bounded, the original Cauchy sequence {an} is also bounded.