My business partner is adamant about who stole the money, what should i do?

A and I own a restaurant together. 2 weeks ago some money went missing in the safe, and ever since then A have been accusing me to be the thief. Couple of days after we discovered some money was missing, a portion of the stolen money was in the safe. After we discovered the portion of the returned money, A changed his mind about me being a thief and assumed i "borrowed" money and was "planning to put it back". I insist on changing the lock to the safe and set up a security camera, but A refused because 1) the store is low on funds and 2) A believes i won't steal the money again since "i got caught".
According to A, I am the number one suspect because i am not good with managing my money and i'm always broke. To me, just because i have money problems doesn't mean i am a thief.

how do i prove my innocence?

Who else had access to the safe? Find the real culprit (if it wasn't you). ~

set up a hidden video camera, it's not hard. Just put it behind something.

Sounds to me like you should be looking at A as the suspect

I believe you don't have to prove your innocence and that your partner sounds guilty to me. I suggest you end the relationship and run from that person while you can. I would also explain your concerns to a lawyer so that someone is aware of the details to protect your partner from ever blaming you for theft etc. run luv2help has good advice put up a hidden camera to protect yourself if you don't want to leave I think you should leave though. I guarantee your partner is stealing profits too. It is only my opinion.

Mr A might be doing this mischief himself to throw you out from the partnership. As you yourself said that yo are always broke and don't know how to handle money, so it is obvious that he might not be happy with you as a partner and innovating ways to weed you out.

Instead of trying to prove your innocence, ask him for a proof and be a little offensive.

I agree with pretty much everyone else. If you and A are the only ones with the combination, then it was one of the 2 of you. But, if you have trouble trusting A, then he might have given the combination to someone else without your knowledge. Have you given the combination to someone else? Again, if only the 2 of you have the combo (and, you're certain of this), then you know who the culprit is. Either you, and you won't admit it, or A, and he's setting you up.