Practice question · Fill in the blanks
Complete the completeness property of the real numbers.
Every non-empty set of reals that is bounded above has ______.
Word bank: a maximum element · a least upper bound that is itself a real number · an upper bound that is rational · only finitely many elements
Hints
- Recall the set of x with 0 <= x < 5, which is bounded above but has no largest element.
- The property must be exactly what the rationals lack.
Show the answer
Every non-empty set of reals that is bounded above has a least upper bound that is itself a real number.
Why
Completeness guarantees a least upper bound, not a maximum, the set of x below 5 has a supremum but no largest element. The rationals fail this: the rationals whose square is below 2 are bounded above but their least upper bound, the square root of 2, is missing from the rationals.
Practise Real Numbers and the Number Line
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