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Every node in the network has the full blockchain (public listing), if a double spend is attempted then it won't be accepted by the majority of the nodes, and consequently won't be included in the next block in the chain.

Double spending is only an issue if you don't wait for the transaction to be "confirmed" by its inclusion in a block. Some services don't do this, and that's where you begin to have problems. You may end up with someone having a service they didn't actually pay for, for example.

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