The most important move in this problem is , failing which, Black dies. Although Black cannot win this capturing race,
and
bridges under and joins the two groups together. There is ony eye at the corner and one eye at the side, so Black lives.
In the solution diagram, the order of and
is interchangable.
The order of the hane at is wrong. This time,
can descend and now Black dies from shortage of liberties - both approach moves at a and b will put the stones into self-atari, so this is a case of double damezumari.
If , then
or a kills. The reason is left as an reading exercise.