In Descartes's Second Meditation, how does Descartes ultimately prove that he exists?

In this section of his meditation, Descartes is trying to make sense of his relationship to the world in two ways, through his senses (sight, smell, taste, touch, sound) and through his intellect (mind). He uses a piece of wax as an example, first describing the wax's physical properties—what it looks, feels, and tastes like. He then places the wax by a fire, where it melts, changing all of its physical properties. He says he now comes to view the wax in other terms, saying that he can know only that it is "flexible and changeable." He notes that "if my perception of the wax seemed more distinct after it was established not just by sight or touch but by many other considerations, it must be admitted that I now know myself even more distinctly. This is because every consideration whatsoever which contributes to my perception of the wax, or of any other body, cannot but establish even more effectively the nature of my own mind." This gets him to his ultimate conclusion of "I think, therefore I am." Basically, he is saying that the mind's ability to take in information, have opinions, make decisions, have doubts, and so on proves that he is alive. It is having a working mind that takes us beyond just being a "body" with senses.