The first fundamental assumption and solipsism

The first assumption that we must make is that we appear to exist.

We need to be careful using the word "exist" because it describes a status that is independent of the mind.
Oxford dictionary:
exist, to have objective reality or being;
objective, a thing or class of things external to or independent of the mind.
In the following, when we talk about existing it will only be an apparent existance that is relative to our mind.
Our apparent existence may appear to be a trivial thing to assume. Indeed it may appear to be self evident (or obvious) and not an assumption at all. But how would a person argue that it is obvious that he or she exists? Perhaps "I know I exist because I am thinking and I remember things I have thought".
In fact Rene Descartes (1596-1650) said "I think therefore I am" meaning I know I exist because I can think. A similar statement was also made by Saint Augustine (354-430).
But the person has merely shifted the burden from "existing" to "thinking" and "remembering". How would the person then argue that thinking is obvious? Again perhaps "I know I think because I have thoughts". So now how is it obvious to have thoughts? We could go on in this manner, but at some point we will arrive at a statement that we have to assume is true (e.g. "I just know I have thoughts"). This assumption is then the basis for claiming existence, and implies that our apparent existence is an assumed thing.

Without making any further assumptions, the point we have reached is the basis for solipsism, the belief that only the self exists. In solipsism all the information we receive from our senses is accepted without knowing or verifying its source. We might feel a pain sensation whenever we have the sensation of moving towards something we label as a fire, but these are just sensations that reoccur in a consistent manner. We don't actually know where they arise from. This view of the world is an entirely subjective one.

Oxford dictionary:
solipsism, the view or theory that self is the only object of real knowledge or the only thing really existent;
subjective, existing in the mind only, without anything real to correspond to it.
The status of the assumption of our apparent existence, and the knowledge that rests upon it, is revealed by considering its contrary. If we do not exist, even apparently, there is nothing further to discuss. So clearly assuming our apparent existence is essential for any meaningful discussion of any knowledge or belief.