Completeness says that Cauchy sequences must converge.

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__Show requirements__

Concept |
Content |

Cauchy
Sequences |
The sequence members of a Cauchy Sequence eventually become arbitrarily close to each other. |

Supremum and
Infimum of Sets |
Bounded sets always have an supremum and infimum which are generalizations of maximum and minimum. |

Real Numbers |
In a real analysis, the real numbers are the largest number set we need. They satisfy axioms that represent the idea of a number line. |

Bolzano-Weierstrass
Theorem |
Every bounded sequence has at least one converging subsequence. |

__Show consequences__

Concept |
Content |

Cauchy Criterion |
A series converges if its partial sums form a Cauchy sequence. As a consequence, for a convergent series, the underlying sequence necessarily needs to converge to zero. |

Uncountability
of the Reals |
The real numbers cannot be enumerated. |

Bounded Monotonic
Sequences |
If a sequence of real numbers is bounded and monotonic, then it is convergent. |

Now we show that Cauchy sequences in \(\mathbb{K}\) are even convergent:

**Theorem 1**. *Every Cauchy sequence \((a_n)_{n\in\mathbb{N}}\) in \(\mathbb{K}\) converges.*

*Proof:* Every Cauchy sequence is bounded.
Therefore, \((a_n)_{n\in\mathbb{N}}\)
is bounded. By the Theorem of Bolzano-Weierstraß it has a convergent
subsequence \((a_{n_k})_{k\in\mathbb{N}}\). Set \(a:=\lim_{k\rightarrow\infty} a_{n_k}\). For
given \(\varepsilon>0\) there exist
\(N_1,N_2\in\mathbb{N}\) such that
\(|a_{n_k}-a|<\varepsilon/2\) for
all \(k\geq N_1\) and \(|a_n-a_m|<\varepsilon/2\) for all \(n,m\geq N_2\). Thus for \(n\geq N:=\max\{N_1,N_2\}\) holds \(n_n\geq n\geq N\) and \[|a_n-a|\leq|a_n-a_{n_n}+a_{n_n}-a|\leq|a_n-a_{n_n}|+|a_{n_n}-a|<
\varepsilon/2+\varepsilon/2=\varepsilon \ . \qquad\Box\]

Theorem 1 is not true for arbitrary normed \(\mathbb{K}\)-vector spaces. Those normed
\(\mathbb{K}\)-vector spaces \((V,||\cdot||)\) for which every Cauchy
sequence has a limit in \(V\) are
called complete or Banach spaces (in honour of the Polish mathematician
Stefan Banach). Without proof we state that all finite dimensional
normed \(\mathbb{K}\)-vector spaces are
Banach spaces.

The next result concerns the special property of the real numbers
that supremum and infimum are defined for all subsets of the real
numbers. This theorem goes back to Julius
Wilhelm Richard Dedekind (1831–1916). It follows from the
completeness axiom (C):

**Theorem 2** (Dedekind’s Theorem). *Every non-empty
bounded set \(M\subset \mathbb{R}\) has
a supremum and an infimum with \(\sup M,\inf
M\in\mathbb{R}\).*

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