We show that various algorithms, ubiquitous in convex optimization (e.g. proximal-gradient, alternating projections and averaged projections) generate self-contracted sequences {x(k)}(k is an element of N ). As a consequence, a novel universal bound for the length Sigma(k >= 0)parallel to x(k+1)- x(k)parallel to can be deduced. In k Ilk addition, this bound is independent of both the concrete data of the problem (sets, functions) as well as the stepsize involved, and only depends on the dimension of the space.