Yes, G-d exists.
How am I going to prove it? I really can't. I believe it, and fervently, but there is truly no way to prove one way or the other.
I sometimes think of it this way...if you drew an entire world on a two-dimensional piece of paper, and then somehow brought them to "life", you would have full control of their lives. If you choose not to directly interfere, they will question if you exist. After all, from their own senses, they cannot perceive you. They don't see your face, they don't see your body, they don't see how you control their world with a pencil. Thus, they would doubt your existence. They would doubt the existence of a third dimension - all they see is length and width. They don't experience a third dimension. If you cut someone off the paper and pasted him back on elsewhere, from their perspective, they just magically disappeared from one place and appeared in another. If you erase half their town, they see it as a massive natural disaster.
I sometimes think of it the same way - G-d exists in such a way that he cannot be perceived directly by our lesser forms, as it is simply not possible for us to think on that level. Our senses cannot operate on his level, basically. Our spirits may or may not be able to, but I'm not going to get into that as that's a completely different theological argument.