The distance of walking can be represented by a function. Suppose a person is walking at a constant speed, then the growth rate of the distance he travels is a constant number. Since the displacement per unit time are all the same, the graph of this function (distance versus time) will be a straight line. So there is no horizontal asymptotes.
Suppose I’m eating a box of cookies. At the first time I eat 1 cookie, and ½ at the second time. And every time afterwards, I’ll eat half amount of the previous.
The amount of cookies I eat every time is a sequence made by a list of numbers. This sequence converges to 0, because I eat less and less cookies.
The total amount of cookies I eat is a series, which sums up how much I ate. This series converges because it is a geometric series with a ratio of ½. And series ∑(0.5)^n converges to 2.