No Infinitely Large or Small Reals

Real Analysis · Axiom Academy

Is there a biggest number? A smallest positive one? Building the real numbers rigorously forces a blunt question: does the line ever end ? Is there a largest real that nothing exceeds, or a smallest positive real just above 0 ? The answer is no on both counts, and one clean fact settles it — the counting numbers 1, 2, 3, climb without bound. Watch that happen first; then drive it yourself. Pick a real number r — drawn here as a marker on the line. Now release the natural numbers 1, 2, 3, . They march to the right, and one of them lands past r . That is the witness: a natural number n with n r . No matter where the marker sits, the counting numbers eventually overtake it — there is no real they all fall short of. Name a number as big as you dare Slide r anywhere — push it to a million if you like. There is a recipe that always beats it: take n = r + 1 , the next whole number up. It is a natural number, and it lands strictly to the right of r . Try to find an r the recipe can't beat. The witness wins every time — so no real r is the largest, and is not a member of . Flip the question. Slide a tiny positive as close to 0 as you can. The same idea undercuts it: take n = 1/ + 1 , then n is a positive real sitting strictly between 0 and . There is always room below. However small gets, n slips beneath it — so there is no smallest positive real, and no positive infinitesimal. Both moves are the same theorem wearing two hats — the Archimedean property of the real numbers:

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