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 $\mathbb{R}$.

Would you like more details or have any other questions?

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."