Suppose, the distance between city A and city B is 1. Each day, I complete $90\%$ of the residual distance. For example, on day 1, I will travel 0.9, on day 2, I will travel $90\%$ of the left distance to city B which is $0.9+0.9(1-0.9)$. By the same logic, on the third day, the distance past will be $0.9+0.9(1-0.9)+0.9(1-0.9-0.9(1-0.9))$, and so on.
Question
- Will the journey be completed in the finite number of days? How could I prove/disprove it?
- What is the probability that, the total distance travelled up to day $k<\infty$ is less than 1? It means find $\Pr(D_k<1)$?
This is one of Zeno's paradoxes. It essentially highlights a discrepancy between physical reality and an idealised mathematical model of that physical reality. In the idealised geometrical model a sequence of points $p_1, p_2, \ldots$ that progresses towards $B$ making a fractional amount of progress each day will never reach $B$. In physical reality, a person or a vehicle can't make arbitrarily small steps (even if it's driven by a "continuous" engine like an electric motor, there will be latency in the engine and a non-zero time lag while the control system issues commands to the engine).