PROOF 201

Prove.

The ratio of circumference to diameter is a single number

On the Euclidean plane.

What you may use

A1of two convex perimeters, the inner one is shorter
Archimedes' postulate — of two curves joining the same endpoints and convex the same way, the inner one is shorter
A2all circles are similar
corresponding lengths of similar figures are in the same ratio — perimeter ÷ diameter does not depend on the size of the circle
A3in a right triangle a²+b²=c²
Pythagorean theorem
A4a bisector divides the opposite side in the ratio of the two sides
angle bisector theorem
A5a bounded increasing sequence converges
monotone convergence (completeness of the reals)
A6aₙ ≤ x ≤ bₙ, both → L ⟹ x = L
squeeze theorem
A7associativity, commutativity, distributivity, cancellation, square roots
arithmetic of the reals
Ea = b ⟹ same operation on both sides
properties of equality

Nothing outside this list may be cited.

Definition D1

For a circle of radius 1, let Lₙ, Uₙ be the perimeters of the inscribed and circumscribed regular n-gons divided by the diameter 2.

Proof.

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