Analysis of Pawn Endgame from Amateur to IM

Question

I started reading the book "From Amateur to IM". In one of the first chapters an analysis is given of Casablanca's Pawn Endgame.

[StartFlipped "0"]
[fen "6k1/7p/8/8/8/8/6PP/6K1 w - - 0 1"]

Now, the book starts by describing several building blocks for going about to win this position. One of them (if black pushes his pawn to h6) is to reach the following position with Black to move and an then do an outflanking maneuver.

[StartFlipped "0"]
[fen "8/8/5k1p/8/5KPP/8/8/8 b - - 0 1"]

1...Ke6 2.Ke4 Kf6 3.Kd5 Ke7 4.Ke5 Kf7 5.Kf5 Kg7 6.Ke6 Kg6 ( 6...h5!? 7.g5 Kg6 ) 7.h5+ Kg7 8.Ke7 1-0

The line given in the book is printed in bold, but in move 6 the pawn h5 push seems to really kill any winning possibilities. Am I overlooking something? Is the analysis in the book flawed?

If indeed the analysis in the book turns out to be flawed, I would appreciate an alternative strategy to win Casablanca's pawn endgame when h7 is played.

Answer 1

White will be able to force the black King away and make progress.

Something like 8.Kd6 Kf7 9.Ke5 Kg6 10.Ke6 Kg7 11.Kf5 Kf7 12.g6+ Kg8 13.Kg5 Kg7 14.Kxh5

Answer 2

White can take the opposition by playing Kd6. To keep the opposition after 8. Kd6 Black would need to play 8...Kf6 or 8...Kh6, but both those squares are controlled by White's g-pawn, so Black will lose the opposition: 8. Kd6 Kf7 9. Kd7 (White now has the opposition) Kg6 10. Ke6 Kg7 11. Kf5 Kf7 12. g6+ Kg7 13. Kg5 winning the h-pawn.

If Black plays Kf5, Kg4 then the White g-pawn will queen long before Black's pawn will reach h2.