= Solution
The masses $\mu_n(\mathbb R)=\psi_n(0)$ converge and are therefore bounded. Part b and <Tonelli theorem> give
$$
\mu_n(\{|y|\geq\lambda\})
\leq C\lambda\int_0^{1/\lambda}
\bigl(\psi_n(0)-\operatorname{Re}\psi_n(u)\bigr)\,du.
$$
The 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.
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