Completeness Criterion (Metric Spaces)

Theorem
Let $(X, d)$ be a metric space.

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

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

Then $X$ is complete.

Proof
Let $(x_n)_{n\in\N}$, be a Cauchy sequence in $X$.

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


 * CompletenessCriterionProof.png

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

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

Now let $m > M$, Let $n_1,n_2 > N$. We have:

Therefore, $(y_{m,n})_{n \in \N}$ is Cauchy in $A$ for $m > M$, and so converges to some limit $y_n \in X$.