If White enters at , Black closes the door with . Then seems to be the best move for White, but with and Black gets just enough eyespace. After , Black is alive - a and b are miai. (Disclaimer: there may be mistakes in this variation.)
Actually, this problem looks harder than it is. When looking at it from White's standpoint, it is quite easy to find the correct move on the spot.
White can enter Black's EyeSpace from two sides: at a or b. Black has to find a move which protects against both. Let's analyze some failures first.
Simply closing off at is not good. makes miai of a and b.
If Black closes off the other side, White breaks in from the first. Up to , Black makes a (slightly modified) door group which is dead.
here fails to protect against the jump to . His eye space is sufficiently reduced. The rest of the moves are for reference.
If White plays or a, Black protects at and makes such nice shape that it isn't even necessary for Black to respond to . We'll analyze this in Gokyo Shumyo, Section 1, Derived Problem 56.