Fin pulls out a library card from a book that Olivia had checked out in her name. In the old library card/check system the card should have been kept at the library to keep track of who had checked it. It should not have been placed back in the book and then released to check out.
When Fin is selecting the train book in the library, he pulls it from a shelf that contains entirely fiction (mostly James Patterson novels). Fiction and non-fiction books would never be on the same shelf - even in a very small library.
Following the scene where Olivia returns home from the hospital, there is a brief shot of Fin and Olivia watching the sunset from her dock. This shot is in fact taken from an earlier scene where Fin and Olivia had a conversation on the dock as they are wearing the same clothing.