Practice question · Select all that apply
Let S be the set of all rational numbers q with q squared less than 2. Select every TRUE statement.
Hints
- Ask where the boundary of S sits on the number line, and whether that point is rational.
- For any rational in S, you can always find a slightly larger rational still in S.
Show the answer
- C. S is bounded above
- D. S has a least upper bound in the real numbers
- E. The supremum of S is the square root of 2
Why
S is bounded above (by 2, say) and its supremum in the reals is the square root of 2, but that is irrational, so no RATIONAL least upper bound exists. This missing boundary is precisely the gap in the rationals that completeness fills. S also has no largest element, since you can always squeeze in a closer rational.
Practise Real Numbers and the Number Line
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