Convergence in distribution

[shan]

Given the definition:
For real-valued random variables X_n, n\geq1 and X, then X_n\stackrel{D}{\rightarrow}X if for every bounded continuous function g: R \rightarrow R, E_n[g(X_n)]\rightarrow E[g(X)]

I want to prove the continuous mapping theorem:
If X_n\stackrel{D}{\rightarrow}X then h(X_n)\stackrel{D}{\rightarrow}h(X) for any continuous function h: R \rightarrow R
without using Skorokhod's representation theorem.

The theorem makes sense to me intuitively but I'm lost as to how to prove it mathematically.

Edit: apologies for the really bad latex, my browser keeps hanging on the preview/save

 

[shan]

If anyone was interested:

Say h(Y_n) = Z_n, h(Y) = Z

E(g(Z_n)) \rightarrow E(g(Z)) for every g that is bounded and continuous (from definition)

E(f(Y_n)) \rightarrow E(f(Y)) for every f that is bounded and continuous (from definition)

E(g(h(Y_n)) \rightarrow E(g(h(Y)) is true because h is continuous and g o h is also continuous, h is also bounded by g