1. Something exists.
2. Every existing being is either contingent or necessary.
3. There is a possible state of affairs in which no contingent being exists.
4. It is necessarily the case that possible states of affairs are explicable.
5. Hence, a necessary being is possible.
6. Whatever is possibly necessary exists in all possible worlds.
7. Therefore, a necessary being exists.
(1) and (2) are easily granted. I cannot doubt my own existence without first existing to doubt it. As for (2), something can either exist in at least one (but not all) possible worlds (contingent existence), or else exist in all possible worlds (necessary existence). Existence in no possible worlds would yield that a thing is impossible, and therefore necessarily non-existent.
(3) is highly plausible. If we represent the sum total of all contingent beings as C, and if all the members of C can at some time fail to exist, it's reasonable to infer that C itself may at some time fail to exist. By analogy, if every part of a house can possibly fail to exist, then the house as a whole can also fail to exist.
(4) doesn't entail that every state of affairs must have an explanation in the actual world. Rather, a state of affairs is defined as "explicable" so long as it is explained in at least one possible world W. Yet, the only thing that is left to explain ~C in W would be a necessary being. So, if it is possible that ~C obtains and that ~C has an explanation in some possible world, then a necessary being exists in some possible world, in confirmation of (5).
(6) and (7) are necessary inferences under the S5 axiom of modal logic. If something necessary does not exist in some possible world, then it is not necessary at all, but contingent, which is contradictory.
Maydole argues very similarly, here. The key difference (besides the use of S5 in the argument above) is that Maydole narrows the scope of his argument to the possibility of C not obtaining in the past, whereas the argument above is more general.