Notice that the stone can be captured, but...
The proof that Black is dead is that can capture...
... followed by . However, there really is no need to play and immediately, because White will waste two moves. cannot recapture the point without causing ko.
In the previous variation, we see that allowing White to play at will kill the group. What if Black plays at ?
here is also a failure. Black captures the stone...
... and the result is no different from the previous failure sequence.
This discussion is at a higher level than the problem and may be skipped by introductory level players on their first reading.
We now turn the problem around and ask how Black can live. There are two ways, each with their pros and cons, but usually the consensus is that one of them is better.
From the previous discussion, it should be easy to deduce that makes two solid real eyes.
Of course, is a legal move, but captures and lives.
Similarly, if here, makes two eyes.
Now we compare both methods.
For the second way to live, Black has three points of territory at a, b and c. However, the disadvantage is that White has one ko-threat at either b or c. On the other hand, creates a cutting-point at d, which may be used for Black's advantage in the later part of the game.
We can see that the pros of the second method significantly outweighs the cons, so the second method is better. Additionally, this is consistent with the observation that each ko-threat is probably worth 1/6 points, then at the bare minimum, we can say that the second method is 1/3 points better than the first method.
For more discussion, please refer to Ko threat exercise 3.