RHP Christmas Quiz and Caruana's Pencil

@greenpawn34 said

A Christmas Quiz, a Christmas Story.
Puzzles and some Red Hot Pawn games.
Blog Post 671

Am I missing something in puzzle one? I think that Re6+ is not mate the King can evade to f6, that is not covered.

While I think that tranforming the pawn into a queen will yield a chekmate regardless of the King's move.