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]