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I have used Coinbase to send bitcoins to an address. Sadly though, I have sent ALL my bitcoins, which left me with the balance of zero. When I checked on the transaction (which is pending at the moment) I saw it added a fee TO THE TRANSACTION which essentially exceeds my wallet balance. Since I sent ALL my bitcoins, let's assume I sent 1btc (just an example) then when I check on block reader it seems the transaction is higher with the fee, i.e. 1.01 btc. Which essentially means I don't have enough funds. What should I do?

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It sounds to me like Coinbase has paid the fee for you, as a courtesy. They have a policy of doing that:

Using our wallet services to send, receive and store digital currency is free. We even pay the network transaction fees which are included with most digital currency wallets.

My understanding is that Coinbase doesn't maintain an individual set of private keys per customer; rather, it pools all deposited funds among its own private keys, and tracks each customer's balance in its internal records. (More like a bank account than a safe deposit box.) So when you requested BTC 1.00 to be sent, it didn't come from an address of "yours" containing only 1.00 BTC, but from some internal Coinbase address(es) with a potentially much higher balance. Thus there should be plenty of funds available in the inputs to pay the extra BTC 0.01 if Coinbase wants to write the transaction that way, and it sounds like they did.

A transaction fee is implemented by having the inputs of a transaction exceed the outputs. So if the transaction does in fact have a fee of BTC 0.01, in addition to your output of BTC 1.00, then by very definition, it must have inputs of at least BTC 1.01. (There might be other "change" outputs sending whatever is left over back to Coinbase, or making payments requested by other customers.) If the inputs weren't valid, that would be a bug in Coinbase's software, and the block explorer you checked should have detected this and warned you about it.

Everything is fine and the transaction should be confirmed soon. When that happens, the payment is irreversible and you can stop caring about how it happened.

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