non-negative almost surely I have a probability measure P and a non-negative sequence of random variables $(X_n)$ and the limit $X=\lim X_n$ exists P-almost surely. I would like to show that $X\ge0$ P-almost surely.

If $X_n$'s are nonnegative, then $\limsup X_n$ and $\liminf X_n$ are nonnegative. If we are given that $\limsup X_n = \liminf X_n$, then $\limsup X_n = \liminf X_n \ge 0$
Also, remember that for a given $\omega$, $X_n(\omega)$ is a sequence of nonnegative numbers and a convergent sequence of nonnegative numbers converges to a nonnegative number.
