No matter how fast you accelerate, you'll chew up distances at a local(!) rate of less than c, so it's enough to consider the limiting case of light and skip the complications introduced by acceleration. The analogous classical problem is the "ant on a rubber rope". If the endpoint of the rope moves at a constant speed, the ant will be able to reach it, no matter the rope's length. If the endpoint accelerates, the ant might not, and only be able to asymptotically approach some point somewhere on the rope.