When i want to make sure the background is really white, after converting to grayscale (Image -> Mode -> Grayscale):
1. I create a new black layer on top of the image.
2. I set the layer mode to blacken (not darken): everything that's not pure white in the bottom layer appears black.
3. I set the black layer opacity to ~50%
4. I apply the Levels command to the bottom layer (move the white level until most background is white, without losing detail in the picture).
5. I delete (paint white) all remaining speckles in the background (making sure I paint on the bottom layer, not on the black layer).
6. I delete (or hide) the black layer.