Completeness Criterion (Metric Spaces)/Proof 1

Theorem

Let $M = \struct {S, d}$ be a metric space.

Let $A \subseteq S$ be a dense subset.

Suppose that every Cauchy sequence in $A$ converges in $M$.

Then $M$ is complete.

Proof

Let $\sequence {x_n}_{n \mathop \in \N}$ be a Cauchy sequence in $S$.

For each $n$ pick a Cauchy sequence $\sequence {y_{n, m} }_{m \mathop \in \N}$ in $A$ converging to $x_n$ like so:

Let $N \in \N$ be such that $\map d {x_{n_1}, x_{n_2} } < \epsilon / 3$ for all $n_1, n_2 > N$.

Let $M \in \N$ be such that $\map d {y_{n_i, m}, x_{n_i} } < \epsilon / 3$ for all $m > M$ and all $n_1, n_2 > N$.

Let $m > M$.

Let $n_1, n_2 > N$.

We have:

\(\ds \map d {y_{n_1, m}, y_{n_2, m} }\)

\(\le\)

\(\ds \map d {y_{n_1, m}, x_{n_1} } + \map d {x_{n_1}, x_{n_2} } + \map d {x_{n_2}, y_{n_2, m} }\)

two applications of the Triangle Inequality

\(\ds \)

\(<\)

\(\ds \epsilon\)

Therefore $\sequence {y_{m, n} }_{n \mathop \in \N}$ is Cauchy in $A$ for $m > M$.

So $\sequence {y_{m, n} }_{n \mathop \in \N}$ converges to some limit $y_n \in S$.

This needs considerable tedious hard slog to complete it. In particular: Proof to complete To discuss this page in more detail, feel free to use the talk page. When this work has been completed, you may remove this instance of {{Finish}} from the code. If you would welcome a second opinion as to whether your work is correct, add a call to {{Proofread}} the page.

Axiom of Countable Choice

This theorem depends on the Axiom of Countable Choice.

Although not as strong as the Axiom of Choice, the Axiom of Countable Choice is similarly independent of the Zermelo-Fraenkel axioms.

As such, mathematicians are generally convinced of its truth and believe that it should be generally accepted.

The dependence on a certain axiom has to be clarified. In particular: Indicate where ACC is required. To discuss this page in more detail, feel free to use the talk page. When this work has been completed, you may remove this instance of {{AxiomReview}} from the code.