How Do We Prove the Existence of God?
 All things are either Necessary-in-themselves or contingent.  Define "The Universe" as the collection of all contingent things.  It follows that "The Universe" must then also be contingent.  Therefore, "The Universe" requires a cause.  The cause of the Universe is either contingent or Necessary-in-itself.  If the cause is contingent, it is included in "The Universe," and is then itself in need of a cause.  An infinite regress of contingents is impossible.  Thus, the cause of "The Universe" cannot be contingent.  So the cause of "The Universe" must be Necessary-in-itself. End.
 All things are either Necessary-in-themselves or contingent.
Things exist. This is obvious. But it's also obvious that things don't simply pop into existence for no reason. This would be absurd. We tend to think things have explanations. This tendency is called the Principle of Sufficient Reason (PSR).
If we denied this tendency then, for example, we could not properly do scientific inquiry. Science investigates the explanations of phenomenon. Imagine if a group of stubborn scientists who denied metaphysics saw a beam of light in the sky and threw their hands up and said it happened just because it did, and that it has no fundamental explanation. We would call these scientists silly, and tell them to do their job better.
Examples aside, let's turn to the definitions. A thing is contingent if it is in need of an explanation. A thing is Necessary-in-itself if it is a wholly sufficient explanation in of itself and explains all contingent facts- that is to say, its existence is required to explain all other contingent things, and that if it didn't exist, we could not have an explanation of anything after it. God is the Necessary-in-itself. If we can find the Necessary-in-itself through our argument, then we have succeeded.
In different words, we say things have causes. God is the uncaused causer. If we can go from the idea that things have causes to finding the uncaused cause, then we have succeeded in proving God's existence.
 Define "The Universe" as the collection of all contingent things.
We can also call "The Universe" the "Big Collection of Contingent Facts" or the "BCCF."
I chose "The Universe" as a name, because the material Universe as we know it is a contingent thing. Yes- all material existence is contingent.
Here is where things can get tricky. The fact that The Universe itself is changing is proof that The Universe is contingent. The Universe, for example, is in a state X in the beginning, at time T1, then it changes to Y at T2, Z at T3… and so on. X existed and then did not exist. X is a state of affairs that depends on some factors to exist, because it may or may not exist. By definition, this is contingent- X has an explanation as to why it is as such (whatever factors existed as to why The Universe was X at T1). (See also these hadith where the Imam (a) discusses this proof:  )
So whatever the explanation or cause of "The Universe" is, it had better not be material, or changing.
 It follows that "The Universe" must then also be contingent.
This follows trivially from .
 Therefore, "The Universe" requires a cause.
If "The Universe" is contingent, then as we had defined, it is in need of an explanation/cause.
 The cause of the Universe is either contingent or Necessary-in-itself.
As we had already set out in , all things are either contingent or Necessary-in-itself.
 If the cause is contingent, it is included in "The Universe," and is then itself in need of a cause.
This follows trivially again from  and .
 An infinite regress of contingents is impossible.
The simplest intuition for this is to imagine for the sake of argument that there were an infinite regress of causes. If one were to imagine a random point “within” the chain and attempted to go trace to the present, it would be impossible to do so, since the present is infinitely far away from any point within the chain. But this is absurd since we exist in the present now, so an infinite regress of causes could not have been the case. We know that there has to be a finite number of steps from any point we choose in the past in our chain to the present.
 Thus, the cause of "The Universe" cannot be contingent.
If it were contingent, it would result in an infinite regress of causes and leave "The Universe" unexplained. According to  an infinite regress is impossible. So the ultimate cause cannot be contingent.
 So the cause of "The Universe" must be Necessary-in-itself.
This follows trivially from . If the cause isn't contingent, it must be Necessary. So the cause is God, the Necessary Being.
And this marks the end of the proof.