Proposition: Convergent Real Sequences Are Cauchy Sequences 

If the real sequence $(a_n)_{n\in\mathbb N}$ is convergent, then it is also a Cauchy sequence.

Notes

• Together with the completeness principle for real numbers, this proposition shows that in the set of real numbers, the set of all convergent sequences is exactly the same as the set of all Cauchy sequences.
• An example of a Cauchy sequence that is not convergent, can be given in the set of rational numbers.
• A similar result can be proven for complex sequences.

| | | | | created: 2020-07-11 15:47:50 | modified: 2020-07-11 16:12:58 | by: bookofproofs | references: [581]

1.Proof: (related to "Convergent Real Sequences Are Cauchy Sequences")