Proving a Division Algorithm for Real Numbers?

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To prove a Division Algorithm for real numbers, one must demonstrate that for any real numbers x and α, there exist unique integers k and real numbers δ such that x = kα + δ, with 0 ≤ δ < α. The uniqueness can be shown by assuming two different representations and deriving a contradiction. To establish existence, it is suggested to consider the sequence na for n = 1, 2, ..., which is unbounded, ensuring that for some n, na exceeds x. By determining the largest integer k such that kα ≤ x, one can express x in the required form. The proof can be simplified by letting k equal the floor of x/α.
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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 &lt; \alpha where k, \delta are unique.
 
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Assume that it is not unique, subtract one representation from the other. The resultant equation is obviously false.
 
Yeah, but how can I show existence?
 
Doom of Doom said:
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
 
or even more straightforward, let k = floor(x/a)
 

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