How could philosopher René Descartes prove his own existence?
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Descartes is analysing all his prior beliefs at the start of the first meditation to find a belief that cannot be doubted and is certain. At the end of the first meditation, he concludes that he is not certain that he is not being deceived about everything by an omnipotent demon, which includes deception of his own existence.
At the start of the second meditation, he is convinced that there is something that can be known with certainty; his own existence. This is one simple way his argument can be formulated:
P1: If I am being deceived by an omnipotent demon about my existence, then I exist (since there must be a target that is being decieved)
P2: If I am not being deceived by an omnipotent demon about my existence, then I exist (since I am not being deceived about my own existence)
P3: I am either being deceived or not being deceived about my existence
C: Therefore I exist (since I exist both if I am being deceived or if I am not being deceived, and these are the only two options)
There are different variations in how you can formulate his argument, but I hope this helps in some way!
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