It is known that
A sequence $(X_n)$ of random variables converges in probability to $X$ if and only if every subsequence $(X_{f(n)})$ has a sub-subsequence ($X_{g(f(n))}$) that converges almost surely to $X$.
Now, assume that for all $\omega\in\Omega$, every subsequence $(X_{f(n)}(\omega))$ has a sub-subsequence ($X_{g(f(n))}(\omega)$) that converges to $X(\omega)$. This should be different from the assertion in the second "if" of the result above, and it should imply that $(X_n)$ converges almost surely to $X$ because $\lim_n X_n(\omega)=X(\omega)$ for each $\omega$. Am I right or doing some mistake?