Proving the sequence $f_{x_n}(x)= (n+1)x^n$ converges in distribution I am preparing for a final exam and just working on sample problems.

Let $X_1,X_2,\dots$ be an infinite sequence of continuous random variables such that $f_{x_n}(x)= (n+1)x^n$ for $0<x<1$ and $0$ otherwise.
Show that $\{X_n\}$ converges in distribution to the degenerate r.v $X$ where $P(X=1)=1$.

Progress: I have found the cumulative distribution for both $F_x$ and $F_{x_n}$:

*

*$F_{x_n} = x^n$ for $0<x<1$, $1$ for $x\ge 1$, and $0$ for $x<0$

*$F_x(x)= 1$ for $x\ge 0 $ and $0$ when $x<0$.

I don't know how to prove it from there on.

Hint: A sequence of rvs converge in distribution is the sequence of corresponding characteristic functions converge for points where the function sequence is continuous. This is Levy's continuity theorem.
So find the characteristic function sequence and show its convergence.
