Hello, sup is the supremum and max is the maximum. A bounded sequence must have an supremum, but it may not have a maximum. For example, an= 1- 1/n, this sequence has an supremum = 1, but it can't get a maximum. But if a series can get the maximum value, that is, the maximum value max exists, then the supremum of this series is equal to the maximum value.

Similarly, bounded functions on intervals must have supremum or maximum. Similarly, for example, y= 1- 1/x, this function has sup y= 1 on [1, positive infinity], but it can't get the maximum value anyway. Similarly, if a function can get the maximum value in a certain interval, that is, max exists, then the supremum of the function in this interval is equal to the maximum value.

Take it if you are satisfied.