Is there any FORMAL proof to the infinite monkey theorem?

I'm currently studiying probability and I wonder if there is an actuall formal proof to this "meme" theorem. Now, please, don't turn this into a discussion about monkeys and typewriters. I'm interested in the juice of the question.

The way I interpret this theorem is that it's not a theorem, but an example of a theorem that would go as follows: "If an experiment that yields a random result is repeated an infinite number of times, any result that has a chance higher than zero to happen will happen at least once".

It's kind of intuitive that this is true, but I haven't learned enough of the subject to form a proof for it myself. The main issue I ran into is that the sample space is not necessarily countable, so the math gets pretty weird.

Every time I search for it online everyone just discusses about how an actual monkey would play with a typewriter.

Is there a known proof that this theorem is true?