The long-run average costs for a firm will never be higher than the short-run average costs, no matter how much output has been produced."

The above statement is true.

It is because in the long run, costs can be minimized since all the factors are variable. In the short run, the fixed factors cannot be changed. Thus, output cannot be changed much in the short run as compared to long run.

The short run average cost is equal to the long run average cost at one point only which happens to be the point of output maximization in the short run. At this point, the variable factors available in the short run are fully and efficiently used and yield the maximum output possible at minimum cost possible in the short run. At all other points except the short run optimum point, the long run average cost of a firm is always less than the short run average cost.