# Pointwise convergence vs weak convergence

I just have a short question:

I’ve prove the fact that given densities fn(x)f_n(x) of the tnt_n distribution, then they converge pointwise to the density Φ(x)\Phi(x) of a standard normal distributed random variable, i.e.
limnfn(x)=Φ(x)\lim_n f_n(x) = \Phi(x)

How can I show the weak convergence t_n \rightarrow ^D N(0,1)t_n \rightarrow ^D N(0,1)? I know, that pointwise convergence is stronger, so one implies the other (pointwise convergence is for every x, weak convergence for every continuity point). But I don’t know how to write it down or prove it properly. Any ideas?

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