Solution

For however many queens are on the board, a box can be drawn around them to include all. Any queen in the corner position can attack no more than 3 queens, so we only need account for situations where the corners are unoccupied. Queens on the side but not in the corner can attack 5 other queens in all directions radiating into the box. But there is always one side queen closest to a corner, which must be unoccupied. That queen can attack only 4 other queens.

Solution

The way Vine Smith words it is sufficient. My solution is more wordy and laborius, but also works. 
 
1) It's obvious that the corners of the board cannot be occupied, since a queen there would attack no more than three other queens. 
 
2) Next, it can be proven that no edge square adjacent to the corner could be occupied, since a queen there would not attack one in the corner, and there are only four other directions to attack. 
 
3) Repeat step #2 for each of the 8 edge squares that are two away from the corner (like c1 or a3) and then the 8 edge square that three away from the corner. Then you are left with at most a 6x6 square that excludes all of the edges with "a few", but still unknown number of queens. 
 
4) Repeat steps 1,2 and 3 for the 6x6 square, which leaves you with only with the 4x4 square in the middle of the board. 
 
5) Repeat steps 1 and 2 and the your left with only the 2x2 square in the middle of the board, and there are only 4 queens left, and none of them can attack more than 4. 
 
6) Repeat steps 1 and 2 for the 4x4 square 
 
Vine Smith's solution is essentially the same, but worded more efficiently and eloquently.