According to my book, the limit sup s_n is always less then or equal to the limit s_n

I don't understand this. If it is the sup, i expect that it is always GREATER than or equal to limit s_n. Furthermore, what bothers me is that the sequence V_N decreases (or is equal to it's previous term), where V_N = sup{ s_n : where n > N }. BUT WHY? Should the V_N not be all the same? because it doesn't matter if n> 20 or n > 100 or n > 100000, the sup is the least upper bound, so whatever the top is for n > 20, should be the same for n > 100000 because 10000 > 20 anyway. In the case of the sequence U_N for the inf, i can understand why U_N increases as N increases because your interval changes. But how does the interval change for V_N ?? The upper bound for n > 20 seems intuitively to be the same for the upper bound where n > 21 say. PLease help.