Descartes came across a similar problem in his ‘meditations.’ There also similar paralells in Augustine and also Asian Philosophy - what is it that thinks? What is my nature?
To say something ‘exists’ is to make a certain statement both about an object and it’s nature. To say ‘Prove’ demands a rational argument. ‘I’ suggests the thinking ego, yourself, or your identity as a body in space-time. ‘Exist’ is a predicate you apply to an object to determine something about it which obviously cannot be true and false at once (i.e. you can’t exist and not-exist at once).
There are two ways we can approach this; we can assert that our perceptions are accurate and correspond to reality, or that our senses are somewhat flawed and don’t give us an accurate picture of reality. If the first way is true, then our existence is fairly self-evident, but if not (a logical argument can be given against it) then we need another approach.
Skeptics and sophists have posed some very good arguments against the existence of the ‘I’, as have some Asian Philosophers or mystics (such as the writers of the Upanishads or the Buddha). Descartes himself actually doesn’t start by saying ‘I think therefore I am’ but instead with the skeptical method which doubts everything including his own existence, only *after * which he comes to the conclusion he is a thinking essence or being, and only this property cannot be taken from him by doubt, as to do so would be to basically cease to exist.
There are a number of objections though that can (and were) raised against Descartes’ arguments, but it is helpful to consider ‘Do I exist’ and what this statement entails logically.