Normed Division Ring Operations are Continuous/Addition

Theorem
Let $\struct {R, \norm{\,\cdot\,}}$ be a normed division ring.

Let $d$ be the metric induced by the norm $\norm{\,\cdot\,}$.

Let $\tau$ be the topology induced by the metric $d$.

Let $\tau \times \tau$ be the product topology on $R \times R$.

Then the mapping:
 * $\phi : \struct {R \times R, \tau \times \tau} \to \struct{R,\tau} : \phi \tuple {x,y} = x + y$

is continuous.

Proof
Let $x_0, y_0 \in R$.

Let $\epsilon \gt 0$ be given.

Let $x,y \in R$ such that:
 * $d \tuple {x,x_0} \lt \dfrac \epsilon 2$
 * $d \tuple {y,y_0} \lt \dfrac \epsilon 2$

Then:

By the definition of continuity then the function:
 * $\phi : R \times R \to R: \phi \tuple {x,y} = x + y$

is continuous.