Consider the following version of the ultimatum bargaining game: Player 1 has the first move. He can choose how to divide a cake of size normalised to 1 and offer player 2 a part of this cake, say x, which is an element of the closed interval [0,1]. Player 2 gets to know player 1's offer and can accept the offer or reject it. If he accepts, he gets x and player 1 gets 1-x. If he rejects, both player get nothing.

My questions:
1. What is the Nash equilibrium of this game?
2. Is this Nash equilibrium subgame perfect*? Why (not)?

*A Nash equilibrium for an extensive form game (an extensive form with perfect information is basically a rooted tree with a partition of the set of moves) is subgame perfect if it induces a Nash equilibrium in every subgame.