Jack convinces Kate that detonating the bomb will save Claire

As the drill reaches the pocket of energy at the Swan Station site, Jack is looking at it happen from a distance. Kate walks up and sees him covered in blood and asks what happened. Jack says "My talk with Sawyer didn't go so good." Kate asks if he remembers when she sewed him up when they first crashed on the island. Jack says that it seems like a million years ago. Kate says "Or 30 years from now."

Jack asks why she made him promise to never ask about Aaron. She says that she was really angry with Jack for making them come back to island. She continues saying that she came back so Aaron could be where he belongs, with Claire. Jack says that if detonating the bomb works, Claire will never come to the island. Kate points out that Claire was going to give Aaron up for adoption and Jack says "Well, you don't know what she would do. And whatever it is, it would be her choice." He continues saying that this is the only way to save Claire and he says that nothing in his life has ever felt so right. He says "And, I just need you to believe that."

At that moment, an alarm sounds and steam rushes out of the hole being drilled at the Swan. Radzinsky screams "Keep the drill going!" Jack says "It's about to happen." He then turns to Kate and asks "Are you with me on this?." Kate pauses for a moment and then answers yes.

So this is a classic Jack and Kate moment. Jack tries to convince Kate that his plan is the right one and Kate eventually agrees. It is quite convincing with Jack says "Nothing in my life has ever felt so right."