I am currently going through the proof of the existence of a solution of the SDE
\begin{align} dX_t = bdt + \sigma dB_t \end{align}
where $B_t$ is a Brownian motion wrt a filtration $\{\mathcal{F}_t\}_{t\in[0,T]}$. Of course $b$ and $\sigma$ satisfies the linear growth bound and Lipschitz conditions, and they are both $\mathcal{F}_t$-adapted. We consider the Picard iterations \begin{align} \begin{cases} X_t^0 &= X_0 \\ X_t^{n+1} &= X_0 + \int_0^tb(s,X_s^n)ds + \int_0^t \sigma(s,X_s^n)dB_s \end{cases} \end{align} and the growth conditions to show that $X_t^n$ converges almost surely uniformly on $[0,T]$ to some process, say $Y_t$. In order for the limit process $Y_t$ to be a solution we need that the equation
\begin{align} Y_t = X_0 + \int_0^tb(s,Y_s)ds + \int_0^t \sigma(s,Y_s)dB_s \end{align}
is satisfied almost surely. We already have that
\begin{align} Y_t = X_0 + \lim\limits_{n\rightarrow\infty}\int_0^tb(s,X_s^n)ds + \lim\limits_{n\rightarrow\infty}\int_0^t \sigma(s,X_s^n)dB_s \end{align}
so it remains to show that
\begin{align} \lim\limits_{n\rightarrow\infty}\int_0^tb(s,X_s^n) - b(s,Y_s) ds &= 0 \\ \lim\limits_{n\rightarrow\infty}\int_0^t \sigma(s,X_s^n) - \sigma(s,Y_s)dB_s &= 0 \end{align}
almost surely. The first of these results follow for instance by the Lebesgue dominated convergence theorem, but I struggle with how to show the other one. Is there some convergence theorem for the Ito integral I can use?
Observations:
1) $\sigma(s,x)$ is continuous (in the second argument) by the Lipschitz condition.
2) By the continuity we have the almost sure limit $\lim\limits_{n\rightarrow\infty} \sigma(s,X_s^n) = \sigma(s,X_s)$.
3) The process $\int_0^t \sigma(s,X_s^n) - \sigma(s,X_s)dB_s$ is a martingale wrt $\mathcal{F}_t$ for each $n$.
Some of observations 1) - 3) might be useful to answer my question.