This is asserted in Exercise 1.15 of Joe Harris's algebraic geometry book (Algebraic Geometry: A First Course, Pg. 11 in my copy). This result struck my fancy but I'm unable to solve it myself or find references to it elsewhere. The closest reference I've found is the following document, which asserts something weaker: http://www.mast.queensu.ca/~tony/kn+1.ps Does anyone know of any references or see a solution off the top of their head?
(A rational normal curve is any curve equivalent to the Veronese image $v_n(\mathbb{P^1}) \subset \mathbb{P}^n$.)