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Which of the following is a true statement in a normed space?
  1. Every Cauchy sequence is convergent
  2. Every convergent sequence is Cauchy
  3. Every bounded sequence is Cauchy
  4. None of these
Explanation
  • In any normed space (which is a metric space), every convergent sequence must be Cauchy.
  • This is because, by definition, if a sequence converges to a limit, its terms eventually become arbitrarily close to that limit—and hence to each other.

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