Fréchet Space (Functional Analysis) is Complete Metric Space

Theorem
Let $\struct {\R^\omega, d}$ be the Fréchet space on $\R^\omega$.

Then $\struct {\R^\omega, d}$ is a complete metric space.

Proof
From Fréchet Space (Functional Analysis) is Metric Space, $\struct {\R^\omega, d}$ is a metric space.

It remains to be demonstrated that $\struct {\R^\omega, d}$ is complete.