Easy way to get Sonic's Bracelet
When you get Sonic's Light Speed Shoes, go to the hotel, and jump right above the man at the desk on the a ledge there will be a door. Go right next to the left side of the door, and start charging up. The beams of the light that go into Sonic when he's charging will hit the switch that opens the door and now you can go in easily.
NOTE: This might not work the first time so keep trying.