Showing that a stochastic integral is a Martingale in $M_2^c$

Question

Suppose $\sigma$ is a real valued, bounded and continuous function and $(X_t)_{t\in\mathbb{R}^+}$ a stochastic process, B a standard Brownian motion. I want to show, that the Integral:$$\int_0^t\sigma(X_s)dB_s$$ is in $M_2^c$. I know this to be true for simple functions and my idea was just to use $\sigma(X_t)^\pi=\sum_{k=1}^{n}\sigma(X_k)1_({s_{k-1},s_k]} $ as a simple function and $\pi$ as a partition, which I will let get finer and finer in the limit to use dominated convergence, which is possible because $\sigma$ is bounded and continuous to show the property for the general case. Is my Idea feasible?