I would like to solve $\int_s^t dW(s)$ without using Ito's rule. I'm currently thinking about this but I'm still worried this is not fully correct..
\begin{align} I &= \int_s ^t dW(s) = \int_0 ^t dW(s) - \int_0 ^s dW(s) \\ &= \lim_{ N \rightarrow \infty} \sum_{j=0}^{N-1} \Big[W(t_{j+1}) - W(t_j)) - (W(s_{j+1})-W(s_j)\Big] \\ &= \lim_{ N \rightarrow \infty} W(t_N) - W(s_N) \end{align}
However, as $N \rightarrow \infty$, we have for $T = n\,\Delta t$, that $\Delta t \rightarrow 0$, hence this means that $I = W(t) - W(s)$.
It feels rather logical but I'm not really sure. Could someone tell me if this is right?