Analysis of Pawn Endgame from Amateur to IM 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.
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