A cauchy sequence of real numbers always _ fill the blanks

A Cauchy sequence of real numbers always **converges to a limit** in the real numbers. This is because the real numbers form a complete metric space, meaning every Cauchy sequence in \( \mathbb{R} \) converges to a point in \( \mathbb{R} \).

In summary:
- A Cauchy sequence of real numbers always **converges** in \( \mathbb{R} \).

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Here are 5 related questions to explore further:
1. What is the definition of a Cauchy sequence?
2. How does completeness of \( \mathbb{R} \) relate to Cauchy sequences?
3. Can a Cauchy sequence in a metric space fail to converge?
4. How can we prove that every convergent sequence is Cauchy?
5. What is an example of a non-Cauchy sequence?

**Tip:** In any metric space, if every Cauchy sequence converges, the space is called "complete."

A Cauchy sequence of real numbers always _ fill the blanks

Explore the concept of Cauchy sequences and their convergence in the real number system. Learn why real numbers form a complete metric space, ensuring that every Cauchy sequence converges to a limit. Suitable for undergraduate students studying real analysis.

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