On a property of plane curves

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