Proving a Division Algorithm for Real Numbers?

[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.

 

[mathman]

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?

 

[mathman]

Yeah, but how can I show existence?

The sequence na for n=1,2,... is unbounded. Therefore for some n, na>x. Find lowest bound, subtract 1 and you will have k. ka<=x, (k+1)a>x, so x-ka(remainder)<a

 

[LukeD]

or even more straightforward, let k = floor(x/a)