Proof of "Paper Folding Theorem"

"We begin by folding a single piece of paper in two. The fold forms a half-circle of radius t, which has perimiter πt. Therefore, πt units of the paper are used in the fold itself. A piece of paper shorter than this cannot be folded even once, because there is not enough paper to form the fold. After folding, the result is a two-layer piece of paper, with total thickness 2t."