This is a cross-post of [this](https://math.stackexchange.com/q/2801027/64809) and [this](https://math.stackexchange.com/q/2800921/64809) questions from math.stackexchange.com since I have not received any response there. I would like to seek help here. 


Suppose $x(t,\omega): [0,T]\times\Omega\rightarrow \mathbf R$ is a random variable on a probability space $\Omega$. 


1. Sequence $(t\_k<2^{-k})\_{k=1}^\infty \implies \lim\_\limits{i\rightarrow\infty}x(t\_i,\omega)\rightarrow0$ for almost all $\omega\in\Omega$. Does this imply $\lim\_\limits{t\rightarrow0}x(t,\omega)=0$ for almost all $\omega\in\Omega$? What if we assume $\lim\_\limits{t\rightarrow0}x(t,\omega)=0$ in probability?
2. For every sequence $(t\_k)\_{k=1}^\infty\rightarrow 0$, $\exists$ a subsequence $(t\_{k\_i})\_{i=1}^\infty$ independent of $\omega$ $\ni\big(\lim\_\limits{i\rightarrow\infty}x(t\_i,\omega)\rightarrow0$ for almost all $\omega\in\Omega\big)$. Does this imply $\lim\_\limits{t\rightarrow0}x(t,\omega)=0$ for almost all $\omega\in\Omega$? What if we assume $\lim\_\limits{t\rightarrow0}x(t,\omega)=0$ in probability?



Does sequence almost sure convergence imply almost sure convergence?
No. A counterexample for all of your questions is as follows. Let $\Omega$ be $[0,1)$, with probability measure $\mathbb P$ being Lebesgue measure. Set $x(t,\omega)=1$ if the fractional part of $1/t$ is $\omega$ and 0 otherwise. 


This is a version of the standard example satisfying convergence in probability, but not pointwise.


