Norm is Complete Iff Equivalent Norm is Complete
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Theorem
Let $R$ be a division ring.
Let $\norm {\,\cdot\,}_1$ and $\norm {\,\cdot\,}_2$ be equivalent norms on $R$.
Then:
- $\struct {R,\norm {\,\cdot\,}_1}$ is complete if and only if $\struct {R,\norm {\,\cdot\,}_2}$ is complete.
Proof
By Cauchy Sequence Equivalence, for all sequences $\sequence {x_n}$ in $R$:
- $\sequence {x_n}$ is a Cauchy sequence in $\norm{\,\cdot\,}_1$ if and only if $\sequence {x_n}$ is a Cauchy sequence in $\norm{\,\cdot\,}_2$.
By Convergent Equivalence, for all sequences $\sequence {x_n}$ in $R$:
- $\sequence {x_n}$ converges in $\norm{\,\cdot\,}_1$ if and only if $\sequence {x_n}$ converges in $\norm{\,\cdot\,}_2$.
Hence:
- every Cauchy sequence in $\norm{\,\cdot\,}_1$ converges in $\norm{\,\cdot\,}_1$ if and only if every Cauchy sequence in $\norm{\,\cdot\,}_2$ converges in $\norm{\,\cdot\,}_2$.
The result follows.
$\blacksquare$