In the book of Bridson and Haefliger it is said that 'it follows easily' from what they proved before. Does anyone know of a rigorous proof that CAT(0) spaces are contractible?
Cat(0) spaces have unique geodesics between points. Fix a base point, and along each geodesic out from that point, pull everything inwards. This is well-defined because geodesics are unique, and is continuous if you use the same map on each geodesic.
i can provide more details as needed.