I don't need to answer anymore questions, you have the jist of it
The three were on a ship at sea when a storm hit.  They were the only three survivors on the only life raft that wasn't destroyed as the ship sank.  They were at sea for days when they made a pact that they would each remove an arm so that they could eat, as they had only water in the rations locker below a raft bench.  The other two went first, as the doctor was the only one who could do it safely.  Before the doctor's turn came, they were rescued.  The doctor, being a man of his word, later amputated his arm and sent it to the others to show that he kept his promise.