as Murch says, you could search a transaction on a blockchain explorer (eg. blockexplorer.com, blockchain.info, ...).
Searching a specific transaction could be according of "transaction hash", "sender address" or "receiver address" of that transaction. It's trivial that if you search by transaction hash, with a high probability a unique result will be show to you, otherwise searching with addresses show all transactions of that address and you must find the proper one that represents your transaction.
After finding your transaction you must see it's status ("unconfirmed" or "n Confirmations") and then you could prove your assertion by presenting this link of blockchain explorer to anybody.
Yes they are public. You should be able to send the transaction ID where you sent them funds and you can verify them. – morsecoder – 2016-03-05T15:06:58.937