On a property of plane curves

Let γ:[0,1]→2[0,1] be a continuous curve such that γ(0)=(0,0), γ(1)=(1,1), and γ(t)∈2(0,1) for all t∈(0,1). We prove that, for each n∈N, there exists a sequence of points Ai, 0⩽i⩽n+1, on γ such that A0=(0,0), An+1=(1,1), and the sequences and , 0⩽i⩽n, are positive and the same up to order, where π1, π2 are projections on the axes.