i need help in the following.
If y_n converges to 0 in probability and F_n is a sequence of functions, then F_n(x+y_n) converges to F_n(x) in probability or the probability that the distance between F_n(x+y_n) and F_n(x) is greater than any positive value is approaching 0 as n approaches infinity.
if y_n is approaching 0, then it almost looks kind of trivial to me that F_n(x) and F_n(x+y_n) are getting closer to each others as n approaches infinity. but i dont know how to show it. the problem is that the index n in F_n.
these F_n are quantile functions which is continuous and differentiable. but i am not sure if this information helps.
If you have any idea on this, could you help me please?
thank you.