Assume that the sequence of stochastic processes $(X_t^n)_{0 \le t \le T}$ converges to $X_t$ uniformly in probability, i.e.
$$\lim_{n \to \infty} P(\sup_{t \le T} |X_t^n - X_t|>\epsilon)=0 \; \forall \epsilon>0, T>0.$$
How do we show that in this case, for any $f \in C_b^2$, i.e. such that $$\Vert f \Vert_\infty + \Vert f' \Vert_\infty + \Vert f'' \Vert_\infty \le C<\infty,$$ $f''(X_t^n) \to f''(X)$ uniformly in probability as well?
My attempt:$ \{\sup |f''(X_t^n) -f''(X_t)|>\epsilon\} = \{\sup |f''(X_t^n) - f''(X_t)|>\epsilon, \sup |X_t^n - X_t| < \delta\} \cup \{\sup |f''(X_t^n) - f''(X_t)|>\epsilon, \sup |X_t^n - X_t| \ge \delta\}.$
The second set is tends to $0$ by assumption that $X_t^n$ converges to $X_t$ uniformly in probability.
But I am having difficulty dealing with the first set. How could we show that this converges to $0$ as well? I would greatly appreciate some help.