i suggest this approach;
1) show that there is no pooling equlibrium (where they all choose the same number). This is easy enough - if all 3 firms are on the same spot they get 1/3; but if one firm moves one towards the centre, it will get at least 0.5. In the special case where all 3 firms are at the center already; any firm can increase profits by moving one away from the centre .
2) show that where at least one firm is not "on top of" the other two, it can increase its profits by moving slightly. By the same logic above, the firm on the "outside" of the group can increase profits by moving one towards the other two. The same logic would apply if all 3 firms were in different positions; either outside firm can increase profits by moving towards the middle one.
Since there is no equlibrium with all firms in the same spot, or with them separated, there is no equilibrium.