Black needs to make an eye on the left edge in sente, and then proceed to make the other eye near the center. So the descent at is the important first move, followed by making the second eye at .
The hane at is wrong. Black can make a thow-in at before returning to win the capturing race at . So there is only half an eye on the left edge, and White dies.
Making the eye at the upper side at first is wrong, because and can destroy the other eye. Most likely the mistake made by here is assuming that will play at , so can live again by playing at ...