A positive integer A is called a \emph{congruent number} if A is the area of a right-angled triangle with three rational sides. Equivalently, A is a \emph{congruent number} if and only if the congruent number curve y2=x3−A2x has a rational point (x,y)∈Q2 with y≠0. Using a theorem of Fermat, we give an elementary proof for the fact that congruent number curves do not contain rational points of finite order.