Euclidean Space is Banach Space/Proof 1

Theorem

Let $m$ be a positive integer.

Then the Euclidean space $\R^m$, along with the Euclidean norm, forms a Banach space over $\R$.

Proof

The Euclidean space $\R^m$ is a vector space over $\R$.

That the norm axioms are satisfied is proven in Euclidean Space is Normed Vector Space.

Then we have Euclidean Space is Complete Metric Space.

The result follows by the definition of a Banach space.

$\blacksquare$