Flea on the coordinate system

Question

We drop a flea on a point of the coordinate system(with integer coordinates). Due to the dimensions of the flea we can not see it. The flea jumps away every second by one unit (always in the same direction). We can choose a point every second and if the flea is on that point at that perticular second we have caught it. Is it possible to catch the flea in infinitely many steps?

Answer 1

If our strategy is to stay still and always check the same coordinate, then we will have 50% chance of catching the flea. Not ideal.

Perhaps more effective a plan, I go arbitrarily far (approaching *infinity) to the positive (right) side of the number line to check. Then the next second I will pick a spot equally far from 0 on the negative (left) side of the number line, such that the flea is necessarily between these two points. Third, I check the coordinate 1 to the right of my first coordinate, and fourth, I check one further to the left of my second coordinate. In this way I continue, getting further and further from 0, alternating halves of the number line. I am progressing 1 coordinate every 2 seconds in either direction, whereas the the flea is moving twice as fast, moving 1 coordinate every second in only one direction, so eventually (in an infinite amount of time) it will catch up to where I am checking. There comes a point where either I catch the flea, or it is on the coordinate closer to 0 by 1 spot than the coordinate I just checked. I then look back on the other side of the number line as the flea jumps into the space I just checked. Finally, I come back to the side of the number line next to the flea, move 1 coordinate further from 0 just like the flea does, and ultimately capture it.

I have a feeling we have to assert the axiom of choice to solve the problem in this way though, where the axiom of choice is stated as:

$$\forall X[\varnothing\notin X\Rightarrow\exists f:(X\rightarrow\bigcup X)\space\space\forall A\in X(f(A)\in A)]$$

Where the arbitrary set A = {integers (coordinates in 1-D)}, X is the set of sets of the form of A, one copy for each second we spend trying to catch the flea, and f(A) is the specific coordinate we choose to search at any given second. Since we take an infinite amount of time, X is an infinitely large set, and because we are also trying to take exactly one element of each element of X, we must have access to the axiom of choice, otherwise we are not assured the existence of f. The empty set is not an element of X, so the axiom applies.

*All infinities except the one starred are countably large.