Vilenkin's spacetraveler must have been traveling infinitely fast in the past. (What follows is a simplified version of the BGV proof with no math):
• Alexander Vilenkin: “Let us now introduce another observer who is moving relative to the spectators [each of whom is motionless except for their riding the expansion of space]. We shall call him the space traveler. He is moving by inertia, with the engines of his spaceship turned off, and has been doing so for all eternity. As he passes the spectators, they register his velocity.”
Since the spectators are flying apart [i.e. the universe is expanding], the space traveler’s velocity relative to each successive spectator will be smaller than his velocity relative to the preceding one. Suppose, for example, that the space traveler has just zoomed by the Earth at the speed of 100,000 kilometers per hour and is now headed toward a distant galaxy, about a billion light years away. That galaxy is moving away from us at a speed of 20,000 kilometers per second, so when the space traveler catches up with it, the observers there will see him moving at 80,000 kilometers per second.
If the velocity of the space traveler relative to the spectators gets smaller and smaller into the future, then it follows that his velocity should get larger and larger as we follow his history into the past. In the limit, his velocity should get arbitrarily close to the speed of light.” [Many Worlds in One (Hill and Wang, 2006), 40.] This is all relevant because, according to standard Einsteinian interpretations, one cannot travel faster than the speed of light, and certainly one cannot travel at infinite speeds.