Proving the R.V. condition

Hello all, I have a problem that I'm not too clear about.

It is as follows:

Let X be a random variable. Show that for every x E R,

B(x) = { w E Omega: X(w) < x} E F

*Where E = element of; Omega = collection of objects, the sample space; w = a member of Omega, elementary or outcome; and F = a field

My solution begins with setting,

B_n(x) = {x E Omega: X(w) <= x - 1/n}

Then I am not too sure how to go about proving it...

I was thinking along the lines of showing equivalence between B(x) and B_n(x), since B_n(x) is already defined to be in the field...

Please help!