While reading the following post : https://almostsuremath.com/2010/03/03/existence-of-the-stochastic-integral/#scn_intexists_eq3
there is a statement I don't understand. It states here that the boundedness in probability of the set in (3) below is equivalent to the statement that for any sequence $\xi^n$ of bounded predictable processes converging uniformly to zero $\int_0^t \xi^n dX$ converges to zero in probability?
The definition of elementary predictable processes are given below.
Why are these two equivalent?
