At the moment of death, the person doesn't vanish into nothingness. What really happens is that what we call 'consciousness' or 'mind' seprates from the body, and then is 'reborn', i.e. connects to a new body due to craving and attachment, depending on the karma one has created in the last as well as previous lifes.
In fact, we can already notice how consciousness' ability o control the 4 material elements that constitute our body gradually weakens:
If we contemplate the impermanence and out-of-control nature of these elements, we understand it's impossible to hold on to them. The more we try to cling to them, the more we will suffer. Knowing and seeing thus, we abandon attachment to both our own body, and the body of loved ones, and we will not be reborn in a course material body again.
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