assume that S is a set of real numbers that don't have a decimal expansion. If S is not empty, then it must have a least element r according to the well-ordering principle. r can be expressed as the sum of two smaller real numbers n and m. Since n and m are smaller than r, they are not a part of S. So they have a decimal expansion. the sum of their decimal expansion is also a decimal expansion that is equal to r. This is a contradiction. Therefore S is empty.
2026-03-30 16:26:37.1774887997
is my proof of the fact that every real number can be represented by a decimal expansion correct?
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As @nickf points out, any ordering of the reals which makes it well ordered is not the standard ordering. In which case your claim that $r$ must be the sum of two smaller numbers that are representable must be using "smaller" in the sense of this non-standard ordering. And that's not obvious to me; you need to prove it.