So , after posting a large number of questions on this Math.SE and Stats.SE , I have finally compiled a set of alterantive definitions of almost sure convergence. They are
$$X_n\xrightarrow{a.s.} X\\\Longleftrightarrow \Bbb{P}(\omega \ : \ X_n(\omega)\to X(\omega)\text{ as }n\to \infty)=1\\\Longleftrightarrow \lim_{n\to\infty}\Bbb{P}(\omega \ :\ \sup_{k>n}|X_k(\omega)-X(\omega)|>\epsilon)=0\ \ \forall \epsilon>0\\\Longleftrightarrow \Bbb{P}(\cap_{n=1}^{\infty} \cup_{k=n}^{\infty} \{ |X_k - X| > \epsilon \}) = 0\ \ \forall \epsilon>0$$
Are all these correct?
Specifically what happens when $X_n(\omega)\to X$ is replaced be $X_n(\omega)= X$ in the second definition (OR am I makinga fundamental mistake)?