## Sequence and Subsequence

Hello all,

There is always a confusing question in my mind regarding sequence and subsequence, particularly in the field of probability theory and stochastic integration.

Given a sequence $H^{n}$ which converges in probability to $H$, we know that there exists a subsequence $H^{n_{k}}$ converging a.s., suppose now we perform some sort of stochastic integration by using this subsequence, $H^{n_{k}} \cdot X$, and this converges a.s. to $H \cdot X$, so how can we conclude this `limit' $H \cdot X$ with the original sequence $H^{n}$, i.e. is $H \cdot X$ in what sense the limit of $H^{n} \cdot X$? a.s.? some other modes? or no conclusion?

Thanks very much.