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I'am trying to solve problem 633 of the elementary set (black's move):

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It seems to me that after black takes the outside liberty, it's a seki. Or is there a better solution?

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Actually, White is dead after you fill that liberty. He can never attack the Black stones, but Black can capture the stone in the corner. After that there is a ko that will kill White. There's no way for White to win the ko - filling it in leaves the white stones in atari. At the end of the game when White has no more ko threats, Black can take the ko, and then the white group.

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