Will the inchworm reach its end on the steady stretching rubber rope

Ant on a Rubber Rope is otherwise called as Worm or Inchworm mathematics puzzle. The solution of this problem appears to be Paradoxical or Counterintuitive.

This puzzle follows a simple principle,

Consider a meter long rubber rope. Inch worm is started moving in a rope 1 km long at a speed of 1 km per second. At the end of the second, the rope was uniformly stretched by 1km per second. Further the rope get stretched 2km long after 1 second, after 2 second rope is stretched 3km.

Here, the question is Whether the Inchworm ever reach its end? We may consider that it cannot reach its end but the answer is amazingly yes, the inchworm can reach its end!

Whatever the length of the rope and the relative speed of the inchworm and stretching, providing the inchworm's speed and the stretching remain steady, so the inchworm will always be able to reach the end given sufficient time. Once if the inchworm began moving on the rope, the rope gets stretched both in Front and Back of the inchworm, conserving the proportion of the rope already walked by the inchworm and enabling the worm to make continual progress.