How can Galaxies be proven to exist in the universe

and how can one prove the depths of what we call space?

"proof" is something you need to build on here...

So... in our galaxy, we can visually see how the nearer ones seem to shift with our orbit about the sun - it's called 'parallax', when a nearer star seems to shift it's position slightly against the stars that are much further away.

We can measure our relative speed with these stars, and we can also measure our relative speed with respect to external galaxies. If we look at the two speeds, we can deduce that galaxies are, indeed, external to our Milkyway.

As for the depths of space, again we have to rely on what we can see, and use deduction. The further a galaxy is away from us, the faster it seems to be moving. By looking in every direction, and by measuring how far away these galaxies are (and, we can do this through a few different methods - cepheid variables being one of them), we can "see" how big the universe must be in order to contain these galaxies moving away from us in different directions.

You can see galaxies with your naked eye you dumb **** . Andromeda Galaxy is clearly visible even without the aid of a telescope.

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I thik what you are probably trying to ask here is how we can know for sure that galaxies are really distant objects, as we now think, rather than being relatively nearby, in our own neighbourhood, as astronomers used to think.

Assuming that really is what you are asking, there are a few clues that can be pretty decisive.

First of all, there is the negative evidence of parallax. If galaxies were nearby, we would be able to measure their distance by parallax. It works like this. You measure the pisition of the galaxy in (say) February) and then again August. During that time, the earth has moved from one side of its orbit to the other - a distance of almost 200m miles. If the galaxy we are looking at were only a few billion or a few tens of billions of miles away, we would see its position change slightly. That's the same principle as an optical rangefinder uses - or indeed the same principle your brain uses to make a 3-D image of the world from two 2=D images made by eyes a few inches apart on your face.