Last time we talked about checkmating with a king and a major piece against a lone king. King and pawn against king is far harder.
Keep your king in front of your pawn. If your opponent has the opposition, advance your pawn one square.
In the first puzzle after 2 Ke7 is a draw... byebye.
In order for the 1st puzzle to work, the white king has to be both in front and have the opposition. As Steking mentioned, he lost it. It helps if he starts out far enough ahead to "waste" a move with his pawn.
Last time we talked about checkmating with a king and a major piece against a lone king. King and pawn against king is far harder.
Keep your king in front of your pawn. If your opponent has the opposition, advance your pawn one square.