“Direct” argument for Nash equilibrium
If each firm charges the price of c then the other firm can do no better than charge the price of c also (if it raises its price is sells no output, while if it lowers its price is makes a loss), so (c, c) is a Nash equilibrium.
No other pair (p1, p2) is a Nash equilibrium since
if pi < c then the firm whose price is lowest (or either firm, if the prices are the same) can increase its profit (to zero) by raising its price to c
if pi = c and pj > c then firm i is better off increasing its price slightly
if pi ≥ pj > c then firm i can increase its profit by lowering pi to some price between c and pj (e.g. to slightly below pj if D(pj) > 0 and to pm if D(pj) = 0).
Note: to show a pair of actions is not a Nash equilibrium we need only find a better response for one of the players—not necessarily the best response.