# Stochastic calculus question on integration convergence

As a non-math student, I have been struggling studying stochastic calculus. Below is the first question I encounter and I have no map in my mind on how to approach this problem.

Let $X_t$, $Y_t$ be two continuous semimartingales. Given $t\leq 0$, let $\mathcal{P}_n$ be a sequence of finite partitions over $[0,t]$ such that mesh($\mathcal{P}_n$)$\rightarrow$ 0. Show that the limit: \begin{split} \int_0^t X_s\circ dY_s= \lim_{n\rightarrow\infty} \sum_{t_i\in \mathcal{P}_n} \frac{X_{t_{i-1}}+X_{t_i}}{2}(Y_{t_i}-Y_{t_{i-1}}) \end{split}

exists in probability, and the process $t\mapsto \int_0^t X_s\circ dY_s$ is a modification of a continuous semimartingale. In addition, for $f\in C^3(\mathbf{R}^1)$, show that:

\begin{split} f(X_t) = f(X_0)+\int_{0}^{t}f'(X_s)\circ dX_s \end{split}