Convergence in distribution and almost sure convergence

Question

Let $(\Omega, \mathcal{F}, \mathbb{P})$ a probability space and $Z_n, Z$ defined on it and taking values in $\mathbb{R}$, such that $Z_n \to Z$, $\mathbb{P}$-almost surely. Let $g_n, g \colon \mathbb{R} \to \mathbb{R}$ measurable, such that $$ g_n(Z_n) \stackrel{d}{\to} g(Z). $$ Question: Does this imply that $$ g_n(Z_n) \to g(Z) $$ $\mathbb{P}$-almost surely?

Thanks!

Answer 1

No. For example, $Z_n = Z\simeq \mathcal N(0,1)$, $g_n(x) = -x$, $g(x) = x$.