"How do you prove the existence of God?"
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Question:  
How would you answer the question "Prove to me that God exists" ? Thanks
Response: 
The first thing that should be pointed out in your question about proving the existence of God is that there exists an entire sub-field of theology known as "apologetics" which is, theoretically, devoted to answering this and similar questions. I am not specially trained in apologetics, and do not necessarily share its enthusiasms, but I will give the best answer I can to this question, from my own discrete point of view, that is. 
Let me start with an historical example. During the Reformation, John Calvin wrote an extensive, multi-volume treatise known by the English title "Institutes of the Christian Religion". In this work, by many accounts his "magnum opus" (though his commentaries are even more extensive), Calvin sought to prove through the most diligent and scholarly research that the "Church Fathers" of yesteryear, far from being at odds with the doctrines being proclaimed by the reformers, were actually in essential agreement with them. For it was in reality the Roman Catholic Church which had strayed from the principles and practices of the traditional faith. This is a highly polished and incredibly detailed work, an exhaustive and systematic defense of the position of the reformed Church from the standpoint of traditional Church writings. Perhaps it was something that needed to be said at the time; perhaps it needed to be written. But it didn't convince the Roman Catholics that they were wrong; and, on the other hand, those who were convinced from scripture, conscience, and the Holy Spirit of the rightness of breaking away from Rome hardly needed the support of scholars and Church-men from centuries earlier to believe what they had already believed. As I say, there were reasons for this sort of thing at this time, when the reformist movement was only just getting its legs. Once opinions had hardened, however, "Institutes" became mostly a curiosity - I'm not that interested in how Roman Catholic doctrine has strayed, and they are not interested in hearing it at all. 
Your question, especially the way in which you phrase it (i.e., "prove to me that ..."), really does parallel this example to a certain degree. For once a person takes this attitude ("You've got to prove it to me!"), there is really very little hope that such a person is ever going to acknowledge God. Why? Because such people have almost inevitably already faced the question of God and have already rejected Him. 
Apologetics of the past were a relatively simple matter (compared to the discipline today), and very often hinged upon various arguments developed for just this purpose, namely "proving the existence of God". For example, the so-called "ontological, teleological, and anthropological" arguments advanced by philosophers and theologians of the past (see, in particular, Charles Hodges' Systematic Theology), "proving" that God exists logically by examination of the very nature of being, universal structure, and human design respectively. While not necessarily without worth, these arguments, to my mind, all approach the matter from the same, somewhat flawed starting point suggested above. That is to say, they are all designed to "prove it" to someone who is no longer listening, and such a thing is generally an impossibility. Human beings have a long and noted track record for being able to ignore the truth and the facts (about anything) when it suits them to do so for whatever reason. 
Even more to the point here is what I believe to be the biblical assessment of this issue. In my reading of scripture, God created all mankind with an innate capacity for understanding and appreciating Him (cf. Eccl.3:11; Acts 17:27). More than this, He also brings every person (of competent and therefore responsible mentality) to the point of recognizing His existence at some time in their life, for it is His desire for all to know Him and choose for Him (Ezek.18:23; Matt.18:14; Jn.12:47; 1Tim.2:4; 2Tim.2:24-26; 2Pet.3:9). If this is the case (and, as I say, I believe this is the scriptural position), then it is unnecessary for us to "prove it" to anyone. For, in the first place, we can't really prove anything to anybody through our own efforts - it is the Spirit of God who makes these issues clear, and those who will not respond to God's Spirit certainly will not respond to us either, no matter how solid our logic or how persuasive our rhetoric. In the second place, in such cases God has most likely already made the truth of Himself quite clear to the people who say this sort of thing to us. To attempt to "prove God's existence" to unbelievers who have already rejected the idea of having any relationship with God at all (and are now even proudly proclaiming His non-existence) may fall into the category of casting "pearls before swine". For in such cases, the best we can hope for is a mere trampling of our truths underfoot. In the worst case, should our words cut to the quick, we stand to be trampled ourselves! And all for the sake of people whose negativity to God and to our Lord Jesus Christ is evident enough in the very challenge "prove it to me". I am not saying we should not be zealous to witness for our Lord, only that we have been commanded by Him to exercise judgment and prudence when confronted with extreme cases like this. Sometimes a simple question that makes such people think twice is the best approach (e.g., “Isn't the fact that you can open your eyes and see a miracle?”). For such people have known about God - it is simply that they have chosen against Him and His Son and are now seeking to justify their choice.
That all do in fact come to be aware of His existence is clear from scripture:
The heavens recount the glory of God, and the firmament tells of the work of His
hands. One day after another pours forth [His] words, and one night after another
declares [His] knowledge. There is no tongue or culture that cannot understand
their voice (i.e., of the heavens/firmament). Their design has gone out into (i.e.,
"is visible throughout") the entire earth, and their words to the end of the world. 
He has set a tent for the sun within them (i.e., hidden it in the
heavens/firmament's night sky), and from this it goes forth like a [resplendent]
bridegroom from his [wedding] canopy. [The sun] exults to run its course like an
athlete [does]. Its starting line is at one end of the heavens, and its circuit [takes
it] to the ends [of the sky]. And nothing is hidden from its view. 
Psalm 19:1-6 
God's wrath is about to be revealed from heaven upon all ungodliness and
unrighteousness – on men who suppress the truth [about God] in their
unrighteousness. For that which can be known about God [from everyday
experience] is obvious to them, because God has made it obvious. His nature,
though invisible, is nevertheless plainly apparent, and has been since His
foundation of the world, for it may be clearly inferred from this creation of His –
[this is true of] both His eternal power and His divinity – so that they are without
any excuse: they knew about God, but they neither honored Him as God nor
thanked Him. Instead, they gave themselves over to [the] vanity [of this world] in
their speculations, and their senseless hearts were filled with darkness. Claiming to
be wise, they became foolish, for they exchanged the glory of the incorruptible
God for images and likenesses of corruptible men, of birds and beasts and
reptiles.
Romans 1:18-23 
I have seen all the work that God has given Man to occupy himself with. [God]
has made everything beautiful in its [limited] time; but He has also placed [the
notion of] eternity in the hearts of mankind – and [He has done this], moreover,
without Man being able to discover the work which God has done from the
beginning unto the end.
Ecclesiastes 3:10-11
Psalm 19 shows that God has placed innumerable witnesses to Himself throughout His creation which are impossible for mankind to refute (the heavenly bodies being merely the most dramatic). The Romans passage states explicitly that "consciousness of God" is universal, while the Ecclesiastes passage explains that this is because God has placed the spark “of eternity” in the heart of every human being. 
Therefore, those who say they don't believe in God have made the conscious decision to reject truth which at one point they knew in their hearts to be true. That is a sobering and important perspective for Christians to have and hold. Those who claim that this is not the case, therefore, are either liars or have truly hardened their hearts to such an intense degree that they have actually forgotten the moment when they became completely aware of God's existence. Sadly, this is far from uncommon. Like the Pharaoh of the Exodus, many have indeed chosen to use their free will to blot out knowledge of the truth within them rather than to respond to it, in spite of the fact that this truth was made decisively clear to them. 
Prove the existence of God? One might as well set out to prove that the sun rises and sets. Everyone knows that this is true (and how can you possibly convince contrarians who may choose to reject what is so patently obvious?). Yes, everyone knows that "He is" - at least at some point and for some moment, however short. The real question, for those who have determined not to reject this truth, is what to do next, and that question can be summed in the name of names, Jesus Christ. 
You can find out more about this whole issue at the following Ichthys links: 
    Peter's Epistles #11: "Natural and Special Revelation" 
    The Satanic Rebellion: Part 1: Satan's Rebellion and Fall, section I.1, "Before He Created the Universe, God Existed" 

    The Satanic Rebellion: Part 4: Satan's World System, section II.6, "The conscience as a restrainer of satanic influence" 

    E-mail response: "What about the "Heathen" (who never heard the Gospel)?" 

Hope this is of help to you.
Yours in Him in whom all things exist and subsist, our Lord and only Savior, Jesus Christ (Col.1:16-17). 
Bob L.
http://ichthys.com