(adsbygoogle = window.adsbygoogle || []).push({}); 1. The problem statement, all variables and given/known data

Show that:

Let S be a subset of the real numbers such that S is bounded above and below and

if some x and y are in S with x not equal to y, then all numbers between x and y are in S.

then there exist unique numbers a and b in R with a<b such that S is one of the intervals (a,b), [a,b), (a,b], or [a,b].

2. Relevant equations

3. The attempt at a solution

Assume if x and y are elements of S with x not equal to y, then all numbers between x and y are in S and S is bounded above and below.

Thus there exists a M, N such that M is greater than or equal to the maximal element of S and N is smaller than the minimal element of S. Also all elements between x and y are inside (M,N).

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# Open and closed intervals and real numbers

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