Show that convergence in probability for arbitrary random variables may be reduced to convergence in probability for bounded random variables by the transformation X'=arctanX
My attempt
arctan is a differentiable invertible continous function from R onto (-90,90) so bounded. if arctan (Xn) converges in probability to arctan(X) then How can I show Xn converges in probability to X
Thank you
$X_n \to X$ in probability if and only if every subsequence of $(X_n)$ has a further subsequence converging almost surely to $X$. Almost sure convergence is preserved by continuous maps. Since $\tan $ and $\arctan$ are continuous the result follows.