Suppose $x(t,\omega): [0,T]\times\Omega\rightarrow \mathbf R$ is a random variable on a probability space $\Omega$. 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?
Here is a closely related question.
Anthony Quas gave the following counterexample for the above question and the related one linked to in the question.
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.