TL;DR exchanges can prove that:
-they own specific amount of bitcoins -and that your bitcoins on their exchange are included in this amount
These stories are a bit alarming and can have some of us wondering if the exchanges we are using actually have the amount of coins they claim to have...
Besides, I don't think losing the keys is the root cause of many exchanges' solvency issues.
TL;DR exchanges can prove that:
https://bitcointalk.org/index.php?topic=22929.msg5286474#msg...These stories are a bit alarming and can have some of us wondering if the exchanges we are using actually have the amount of coins they claim to have...