An ant starts from a rope of length 1m and with a speed of 0.1m/sec and simultaneous rope expands at a rate 1m/sec. If we find the ratio of end points to ant's position we get a sequence that converges to 1 (check it for yourself) and hence this implies that ant reaches the end point at some large enough time t.
Variation: Instead of starting at 0, the ant starts at 0.1m. Now the ratio of end point to ants position is fixed ie 1/10 and remains same throughout and hence never reaches 1, therefore the ant never reaches the end point. So here is the counter-intution, why come starting ahead be a disadvantage?