Your skin has two layers: The outer layer (epidermis) and lower level (dermis)
You shed skill cells from your epidermis and they are replaced all the time. Your dermis is much more stable and does not undergo this kind of dying and replenishment
Tattoos use a needle to get the ink past your epidermis, into your dermis. If you got tattoo ink and just painted it on you (on the epidermis layer), it would indeed wear off over time
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