Practice question · Multiple choice
When examining why the rational number line fails completeness, why does the bounded collection of fractions with squares below 2 expose an unfilled gap?
Hints
- Name a rational upper bound for the set - 1.5 works. Now try to find the smallest one.
- For any rational upper bound you propose, can you find a smaller one that still works?
Show the answer
B. Since the supremum is √2, which is an irrational number.
Why
Upper bounds are abundant — 1.5, 1.42, 1.415 — and there is no smallest one, because any rational bound can be lowered slightly and still clear the set. Density is not completeness: density says there are no jumps, completeness says no holes where a limit should be. Without it the intermediate value theorem and bounded monotone convergence both fail.
Practise Real Numbers and the Number Line
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