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## Main Question or Discussion Point

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

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

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