Let $W_t$ be one-dimensional Brownian motion, to calculate $\int_0^tW_sdW_s$ by the definition of stochastic integration, one way is to use the integration of $W^{(n)}_t=W_{[nt]/n}$ to approximate $\int_0^tW_sdW_s$.
My question is the definition of stochastic integration require the simple function to be bounded, but $W^{(n)}_t$ is not bounded. How to reconcile it? Thanks very much!