Enlarging the eye space with is the key move, and probably not too hard to come up with. After Black blocks with , White's descent with is tesuji. Whatever Black tries hereafter, he can't reduce White to one eye anymore.
White's follow-up at here is also a mistake. makes miai of a and b to kill White. at c is answered by Black d, creating shortage of liberties. (Note: should not be at b: White blocks at c and has a ko to live.)
Mike Terry: Depending on what is further up the board, White may be able to continue from the previous diagram, resisting stubornly as shown here. After , White will either live in the corner or break out up the left hand side. Black can prevent all this by playing at a, leading to a ko, but in any case this option is clearly inferior for White, compared to living unconditionally as in the proper solution.
demetria: I've only just started studying Go, so I'm having a hard time imagining this. What happens if black plays at A? Doesn't that stop white from enlarging the eye space?