I am currently revising some martingale theory, and was trying out an old past paper question. I haven't come across anything like this before, so was wondering how one would approach it, and also in general if there are any ways of finding the limit of a sequence of martingales.
Question 4
Assume that $B$ is a 1D Brownian motion from $0$. Define $$ \begin{equation}\tag{1} M^f_t=f(B_t)-f(0)-\frac{1}{2}\int_0^tf^{\prime\prime}(B_s)ds \end{equation} $$ Also define $$f_n(x)=\begin{cases}|x|&\text{for }|x|\geq 1/n\\\frac{n}{2}x^2+\frac{1}{2n}&\text{otherwise}\end{cases}$$ For all $n$ you can assume $M^{f_n}$ is a continuous martingale, and that moreover $$\begin{equation}\tag{2}\mathbb{E}\left[|M^f_t|^2\right]=\int_0^t\mathbb{E}\left[|f^\prime(B_s)|^2\right]ds\end{equation} $$
Show that there is a continuous martingale $M$ s.t. for all $t$,
$$\begin{equation} \mathbb{E}\left[\sup_{s\leq > t}|M^{f_n}_s-M_s|^2\right] \to 0\text{ as }n\to\infty\end{equation} $$
I assume the argument is that since the $f_n$ are converging uniformly to $x\mapsto |x|$, then the resulting limit of $M^{f_n}$ is still a continuous martingale, but I am not sure how to make this precise.