Sartre's proof of universal being i think the approach is to understand that there is a guiding method being used which is the "phenomenological" method. It is a case of applying the method of simply describing what it is which appears as an appearance.
Remember we are applying systematic doubt at all points and making no assumptions but that which is not an assumption is that being appears or rather a being, i.e. there is something, a totality, appearing as a fact to me at any particular time, like a cup.
So the first word we can use without any controversy is "it".
What describes it in the first instant? Well that is that it is manifest, i.e. it presents itself as being.
But at the same time there is the flip side of the phenomenon which is that it is an appearance to a subject so we can talk about "modern thought" which is what I am doing to manifest it.
How does one manifest something? Well it has to be by way of an appearance, i.e. something that is arriving and has some sort of content. There is a series of appearances which manifest it and that is exactly the content of your memory, the existent.
It goes on like this until we have constructed our first sentence which i listed before, and we know for sure that it is correct while it appears as a fact, whatever it is.
I know i haven't got this fully correct but would love to have some help in putting this together.