The outline of reasoning goes as follows:
First, the set of pure strategy subgame perfect equilibrium payoff is the largest pure action self generating set.
Second, since utility functions are all assumed to bounded in $\Bbb R$, the closure of the set of pure strategy subgame perfect equilibrium payoff must be compact. So showing the set of pure strategy subgame equilibrium payoff is compact means showing showing the set of pure strategy subgame equilibrium payoff is closed.
Third, any Cauchy sequence of pure strategy subgame equilibrium payoffs corresponds to a sequence of a decomposable pair, i.e. action and a continuation promise. When the action space are all finite, the space of decomposable pairs is a compact subset of finite dimensional Euclidean space. So it must contains a convergent subsequence whose limit corresponds to payoff. By continuity, this payoff must be a pure strategy subgame equilibrium payoff.
But I can*t understand the case that some action space is a continuum. Fix a Cauchy sequence of pure strategy subgame equilibrium payoffs. In the proof, the action space is transformed to a finite partition. Each cell of partition is regarded a action in a new finite action space. This formulation does admit a convergent subsequence. But how to choose