Just something I randomly came up with. Sorry if it is a duplicate.
I don't think it is possible to forcibly mate the king with rook and knight only.
Proof: The only mating position is with the white king in a corner, the rook giving check from an adjacent square and the knight protecting the rook and covering the escape square.
For simplicity let's say Ka1, Rb1, Nc3. There are 7 other equivalent positions to this one.
If you retro-analyze the position the last move must have been with the rook along the b file from somewhere between b3 and b8. For the following analysis it does not matter from where it came.
Then what was white's last move? It certainly was a king move from a2, b2 or b1 to the corner (a1). However, since white was not forced to move the king to the corner but could have moved to a3 or c1 (or to b3 if the rook was standing there), the mate cannot be forced.
[fen "1r6k/8/8/8/8/K7/4n3/8 w - - 0 1"] [StartPly "4"] 1. Ka2 Nc3+ 2. Ka1?? (2. Ka3!) Rb1++
It's likely not a win, assuming White plays rationally. While it is possible to set up a checkmate position (e.g., knight on c3, rook on a2, White king on a1), Black shouldn't be able to force White into this position. The knight and rook can't control enough squares in their immediate vicinity to create a box and successfully push the White king back.
The knight would have to go to c3, and then the king's only move is a1. Then Rb1 or Ra2 mate, the Arabian mate.
But on 1.Nc3+, the king can go to a3. It can't go to the b-file, because we've assumed a rook is on b8. A similar argument holds for if the king starts on b1, with a rook on h2.
Had a king been on c4, 1.Nc3+ Ka3 2.Rb3#. Therefore, a king is necessary.
P.S. The Arabian mate is so-called because it exists from the times of Chaturanga (500 A.D.)! The rook, knight and king have always been able to move the same.