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Prokhorov's theorem says that a sequence of Borel probability measures on $\mathbb{R}$ is relatively compact for weak convergence of probability measures exactly when it is tight: for every $ϵ>0$, some compact $K$ satisfies ${\sup }_n{\mu }_n(\mathbb{R}\setminus K)<ϵ$.

The masses ${\mu }_n(\mathbb{R})={\psi }_n(0)$ converge and are therefore bounded. Part b and Tonelli theorem giveThe integrands are uniformly bounded. By pointwise convergence and the continuity of $\psi$ at zero, the right side can be made uniformly small for all sufficiently large $n$ by taking $\lambda$ large; finitely many remaining measures are individually tight. Thus $({\mu }_n)$ is tight. Applying Prokhorov's theorem after normalizing the masses, or adjoining missing mass at one fixed point, gives a weakly convergent subsequence.