Self-contracted curves have finite length

A curve theta: I -> E in a metric space E equipped with the distance d, where I subset of R is a (possibly unbounded) interval, is called self-contracted, if for any triple of instances of time {t(i)}(i=1)(3) subset of I with t(1) <= t(2) <= t(3) one has d(theta(t(3)), theta(t(2))) <= d(theta(t(3)), theta(t(1))). We prove that if E is a finite-dimensional normed space with an arbitrary norm, the trace of theta is bounded, then theta has finite length, that is, is rectifiable, thus answering positively the question raised in Lemenant's paper ['Rectifiability of non Euclidean planar self-contracted curves', Confluentes Math. 8 (2016) 23-38].