Induction in Geometry

Given: a unit length.

Use only a straightedge (ruler without markings) & a compass.
Prove: we can construct a line segment of √n for all n ∈N.
Proof:
1) n=1 (given).

2) Assume true for n, i.e. can construct √n

3) Construct a right-angled triangle with height = 1, base= √n

=>  hypotenuse  = $latex \sqrt {n+1} $
=> True for n+1

Therefore true for all n ∈N [QED]