Completeness Criterion (Metric Spaces)/Proof 1

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:


 * CompletenessCriterionProof.png

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:

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 $y_n \in S$.