One can believe anything even without proof. But can one rationally believe in something that has been proven?

The word "believe" has very loose meanings in casual, everyday talk. Technically, if you know something to be true, you don't believe it anymore ... you know it.

But people use that word in many ways. I can say I believe that gravity exists, even though I know it does.

The thing about proof is that it is irrefutable, everyday we can, as you say, accept a great multitude of statements without incontrovertible sensory proof- such as 'it is dangerous to use a phone at a petrol station' and, although there is evidence to suggest that a phone may produce a spark, it has not been undeniably proven that this spark is any more dangerous or likely to prove ignition than the static produced by our clothing.

Hence we can only rationally believe something with irrefutable proof for its verity, such as the fact that London is in England. Equally we cannot rationally believe something supported only by evidence but not proof, such as the idea that soft drinks cause cancer. Our irrational belief of unproven statements is somewhat intrinsic to human nature and evident, not only in religion/non religion, but primarily in daily life. If we were, indeed, to stop believing in all that lacked utter proof for its verity, progression would be stunted by our acceptance of only that which is tangible- disabling the worth of either science or religion and rendering us quite unable to formulate any new ideas.

Yes. Charles Darwin believed in his unproven theory of evolution while believing at that time in the proven theory of spontaneous generation. He was such a smart man.

Do you believe in selective breeding as a way to get better bloodlines in livestock?

So do I.

Tithing as described in Numbers is clearly about selective breeding.

Written thousands of years before "Science".

I don't believe in anything, but if something is proven and I see this proof then I believe it: but not believe in it.

No

Belief isn't rational to start with.

You want to be shown?

He that hath my commandments, and keepeth them, he it is that loveth me: and he that loveth me shall be loved of my Father, and I will love him, and will manifest myself to him.

no, because nothing has ever been proven, ever! you can't even prove that you exist.

Nah, it's more in the realm of acceptance rather than belief.

If there was proof it wouldn't be your belief, it would be fact