Bran is going to change history. One of the last scenes in the series will be Robb Stark and Catelyn Stark receiving Rhaegar, the Kingsguard, Prince Aemon Targaryan (Jon) and the rest of the royal family at Winterfell similar to the first season as they march to the Wall to meet the threat of the Others reported to them by Lord Commander Ned Stark who everyone trusts enough to listen to. In the current timeline the world is destroyed.

The Others cannot be stopped due to the fractured situation of the realm and Bran must slowly learn all that he needs to learn about the past and how events added up to bring them to such dire circumstances before actually utilizing his ability to change the past. Bloodraven is lying to him about the "ink being dry" to prevent Bran from acting selfishly and screwing the timeline horribly and possibly creating a similarly apocalyptic result to the Others invading. After Bran learns "everything" that he needs to know, he will identify a key moment in history and make the correct decision on his own to change it for the good of the world rather than his own personal desires. Bran will leave the cave, because history will be changed and he'll never arrive there to begin with in the corrected timeline.

Robert loses the rebellion, Ned takes the black, conceives Bran on his way to the Wall, Robb becomes Lord of Winterfell, Rickon is never born (a Shaggydog story), Rhaegar overthrows his father, and the Others are defeated. It will be a bitter sweet ending because we will see Jon as an entirely different character and the interactions everyone has with each other will be entirely alien. Jaime will no longer be a Kingslayer, Catelyn will show respect to Jon as the crowned prince, as will Alliser Thorne who will have never had to take the black and will instead be marching as a loyalist to the Targaryen force, Robb and Jon will show mutual appreciation but nothing close to the brotherly affection they had, etc. The world will be saved, but the world we've all become attached to will no longer exist. So it's most certainly not the last we've seen of them.