For my master thesis I have to show that a fractional brownian motion is not a semimartingale. So I searched for an easy way to show this and come to the following statement for semimartingales Let $K_S^n=\sum_{k=1}^{k_n}\alpha_k^n \mathbb{1}_{(t_{k-1},t_k]}(s)$ be a simple predictable process, $\alpha^n_k \in F_{t_{k−1}}$ and $0=t_0<t_1\cdots<t_{k_n}=T$. A stochastic process is an semimartingale if and only if for any sequence of predictable simple processes $k_n$ with the property $\sup_{s≤T}|K^n_s|\to0$ in probability we also have $$ \sum_{k=1}^{k_n}\alpha_k^n(X_{t_k}-X_{t_{k-1}})\to 0$$ in probability. This statment was mentioned here Why is a fractional Brownian motion not a semi-martingale?
My problem is that I dont find any source for this theorem. Does anybody knows a book or a valid source where the statement is formulated specific and is equipped with a proof?
Thanks.