Proofing Metropolis Hastings

I need to prove the correctness of the Metropolis Hastings algorithm in a presentation.
Sadly, I am having trouble on where to start and what to actually proof.

So if I understood correctly, my final goal is to show that given a proposal function q, it will eventually converge to my target distribution P. My biggest problem is I, that I am having trouble on where I can see statements as given. For instance, i need the underlying PDF to be stationary, but can I just assume this step? My MCMC needs to be ergodic so i know it is an invariant measure, can I just assume that? I know that by Lemma, if a the proposal function is reversible then my then my distribution is invariant.
I am having trouble patching all this information together.

Furthermore, my statistics courses have been some time ago so I am having a bit of trouble with understanding all the technical terms that fly around in those posts.

I know my question is very broad, but can someone try to give me a simple explanations of each steps a MH proof should have?