proof firstwe show that xn - xn - using the inequality [ [afor any event to this endconsider the distribution function of xf xn (xp[xn xp[xn |xn xep[xn |xn [|xn xep[xn xx xn [|xn xep[ enowin the arguments above we can switch the roles of xn and (there is symmetryto deduce the analogous resultf ( [| xn ep[xn ethereforemaking the switch gives ( [| xn ef xn (xputting it all together givesf ( ep[| xn xn ( [|xn xef ( etaking on both sides yields for any ( lim xn ( ( ensince is continuous at by assumption we can take to conclude that limnf xn (xf (xlp lq secondwe show that xn - xn - for since the function (xxq/ is concave for / jensen' inequality yields( |xp ) / ( |xp (|xp | | in other words( |xn - | ) / ( |xn -xp ) / - proving the statement of the theorem chebyshev' inequality thirdwe show that xn - xn - first note that for any random variable ywe can writee|ye[| {| |>ee[| {| |>ee [|yethereforewe obtain chebyshev' inequalitye| ( [|ye using chebyshev' inequality and xn -xwe can write [|xn xe |xn - henceby definition xn - cpl finallyxn - xn - xn - is proved in examples and finallywe will make use of the following theorem theorem slutsky let (xybe continuous scalar function of vectors and suppose that xn - and - for some finite constant thend (xn - (xc