I think this is not quite right as stated there because the root mean square (quadratic mean) is always positive or 0 while the arithmetic mean can be negative, making it smaller. I guess the inequality only holds for positive numbers.

That's actually one argument for not calling the root mean square a "mean", because a mean should arguably have the property that it is always a number between the largest and smallest value. But the RMS of two negative numbers is positive. (On the other hand, the median would qualify as a mean in this sense, even though it is not a "power mean".)