# Division algorithm for R

Doom of Doom
How might I prove a Division Algorithm for the Real numbers?

That is to say, if $$x, \alpha \in \mathbb{R}$$, then $$x=k \alpha + \delta$$ for some $$k \in \mathbb{Z},$$ $$\delta \in \mathbb{R}$$ with $$0 \leq \delta < \alpha$$ where $$k, \delta$$ are unique.

## Answers and Replies

Assume that it is not unique, subtract one representation from the other. The resultant equation is obviously false.

Doom of Doom
Yeah, but how can I show existence?