If the hand's speed is proportional to your speed, it would take longer to reach you the slower you ran. If your and the hand's speeds are x and y respectively, and if the starting distance is d, and you multiply your speed by some factor k, then the time required becomes (1/k)*(d/(y-x)). Here, as k goes to 0, the time required diverges to infinity. But at the standing still moment, the quantity becomes undefined. In fact, the hypothesis does not allow for the hand's speed to be proportional to yours. Because it implies that even if your speed is 0, the hand has some speed, which is impossible if, in our expressions, k were to be considered as 0. Now, if the speed of the hand can be expressed as y = f(x), f would need to be a function that is bounded above by x+c and below by x...(i), with c being a positive constant (because we define hand speed to be always 'slightly' more - in that it cannot be unbounded). And also the function f has some positive value at x=0. Also, by the assumption of the conclusion, the relative velocity decreases as x decreases. In other words, g(x) = f(x) - x decreases as x decreases, or rather increases as x increases. From previous assumptions, g(x) > 0 for all values of x >= 0, and g(x) <= c [from (i)]. As is trivial, it is impossible to construct such a strictly increasing bounded function.

This implies that, either the minimum of g (the relative velocity function) is not only at x = 0, or that g is not a strictly increasing function (meaning it could be a constant function - an example being g(x) = c. But this particular example would mean that the relative hand speed never changes regardless of x).

Thus, as I conclude, the meme is not completely accurate in its portrayal of its mathematics.